I began my rental on 12/27 (less than 14 days of the year), and received payment for 30 days in December. For cash basis accounting, how should I file?

It depends on how you entered information into the program. if the property was your personal residence before becoming a rental, then you do not report the income because it was rented less than 15 days.  Check this screen and see if you marked this correctly or not. If this wasn't your residence prior to being a rental, you must report the full amount of the income and you won't receive this message. 

As far as for 2023, what was the terms of the agreement for the rent that was paid for December? Did part of that rent cover a portion of Jan 2023?

Rental income is reported as such in the tax year it is paid/recieved. Period.

The 15-day rule does not apply here, because your "intent" is long term rental - not short term.

Now I get the impression that you're a first time landlord. Really can't be sure, as I've learned that sometimes assumptions can come back to bite. But just in case, I'd like to make you aware of a few things.

When dealing with long term residential rental real estate, absolute perfection in that first year is not an option...... it's a must. Even the tiniest of mistakes can (and will) grow exponentially over time. Then when you catch the mistake years down the road (usually in the tax year you sell the property) the cost of fixing it "will" be high. So if you have questions, by all means, please ask.

As you work this through the program there are some screens that in my personal opinion, do not provide you the necessary clarity I think they should. (We all know what opinions are like!). This lack of clarity may lead to you making incorrect selections and/or entering incorrect data without realizing it. In the hopes to help you avoid that, I've provided the below information for your use. Use it or not, as you see fit. But hopefully you find it helpful.

Rental Property Dates & Numbers That Matter.

Date of Conversion - If this was your primary residence or 2^nd home before, then this date is the day AFTER you moved out, or the date you decided to lease the property – whichever is later.
In Service Date - This is the date a renter "could" have moved in. Usually, this date is the day you put the FOR RENT sign in the front yard.
Number of days Rented - the day count for this starts from the first day a renter was contracted to move in, and/or "could" have moved in. That would be your "in service" date or after if you were asked for that. Vacant periods between renters do not count for actual days rented. Please see IRS Publication927 page 17 at https://www.irs.gov/pub/irs-pdf/p527.pdf#en_US_2020_publink1000219175 Read the “Example” in the third column.
Days of Personal Use - This number will be a big fat ZERO. Read the screen. It's asking for the number of days *YOU* lived in the property AFTER you converted it to a rental. I seriously doubt (though it is possible) that you lived in the house (or space, if renting a part of your home) as your primary residence, 2nd home, or any other personal use reasons after you converted it to a rental.
Business Use Percentage. 100%. I'll put that in words so there's no doubt I didn't make a typo here. One Hundred Percent. After you converted this property or space to rental use, it was one hundred percent business use. What you used it for prior to the date of conversion doesn't count.

RENTAL PROPERTY ASSETS, MAINTENANCE/CLEANING/REPAIRS DEFINED

Property Improvement.

Property improvements are expenses you incur that Improve, restore, or otherwise “better” the property. Basically, they retain or add value to the property.

Betterments:
Expenses that may result in a betterment to your property include expenses for fixing a pre-existing defect or condition, enlarging or expanding your property, or increasing the capacity, strength, or quality of your property. An example of a pre-existing condition or defect in this context would be something such as foundation repair (slab jacking) or some other, hidden and costly, anomaly.
Restoration:
Expenses that may be for restoration include expenses for replacing a substantial structural part of your property, repairing damage to your property after you properly adjusted the basis of your property as a result of a casualty loss, or rebuilding your property to a like-new condition.
Adaptation:
Expenses that may be for adaptation include expenses for altering your property to a use that isn’t consistent with the intended ordinary use of your property when you began renting the property. Adding a wheelchair ramp would be an example.

Expenses for these types of costs are entered in the Assets/Depreciation section and depreciated over time. Property improvements can be done at any time after your initial purchase of the property. It does not matter if it was your residence or a rental at the time of the improvement. It still adds value to the property.

To be classified as a property improvement, two criteria need to be met:

1) The improvement must become "a material part of" the property. For example, remodeling the bathroom, new cabinets or appliances in the kitchen. New carpet. Replacing that old Central Air unit.

2) The improvement must retain or add "real" value to the property. In other words, when the property is appraised by a qualified, certified, licensed property appraiser, he will appraise it at a higher value, than he would have without the improvements.

There are rules that allow you to just flat-out expense and deduct some property improvements instead of capitalizing and depreciating them, if the total cost of the improvement was less than $2,500. It’s referred to as “safe harbor di-minimis” But depending on the specific situation, this may or may not be beneficial. Just be aware that not every property improvement that cost less than $2,500 qualifies for this. If this interest you, the rules can get complex. So a good place to start reading is on the IRS website at https://www.irs.gov/businesses/small-businesses-self-employed/tangible-property-final-regulations. The stuff on di-minimis starts about one page down.

Cleaning & Maintenance

Those expenses incurred to maintain the rental property and its assets in the usable condition the property and/or asset was designed and intended for. Routine cleaning and maintenance expenses are only deductible if they are incurred while the property is classified as a rental. Cleaning and maintenance expenses incurred in the process of preparing the property for rent for the very first time are not deductible.

Repair

Those expenses incurred to return the property or its assets to the same usable condition they were in, prior to the event that caused the property or asset to be unusable. Repair expenses incurred are only deductible if incurred while the property is classified as a rental. Repair costs incurred in the process of preparing the property for rent for the very first time are not deductible.

Additional clarifications: Painting a room does not qualify as a property improvement. While the paint does become “a material part of” the property, from the perspective of a property appraiser, it doesn’t add “real value” to the property.

However, when you do something like convert the garage into a 3^rd bedroom for example, making a 2-bedroom house into a 3-bedroom house adds “real value”. Of course, when you convert the garage to a bedroom, you’re going to paint it. But you will include the cost of painting as a part of the property improvement – not an expense separate from it.

Reporting the Sale of Rental Property

If you qualify for the "lived in 2 of last 5 years" capital gains exclusion, then when prompted you WILL indicate that this sale DOES INCLUDE the sale of your main home. For AD MIL personnel who don't qualify because of PCS orders, select this option anyway, because you "MIGHT" qualify for at last a partial exclusion.

Start working through Rental & Royalty Income (SCH E) "AS IF" you did not sell the property. One of the screens near the start will have a selection on it for "I sold or otherwise disposed of this property in  2021". Select it. After you select the "I sold or otherwise disposed of this property in 2021" you continue working it through "as if" you still own it. When you come to the summary screen you will enter all of your rental income and expenses, even if it's zero. Then you MUST work through the "Sale of Property/Depreciation" section. You must work through each individual asset one at a time to report its disposition (in your case, all your rental assets were sold).

Understand that if more than the property itself is listed in your assets list, then you need to allocate your sales price across all of your assets.  You will only allocate the structure sales price; you will NOT allocate the land sales price, since the land is not a depreciable asset.  Then if you sold this rental at a gain, you must show a gain on all assets, even if that gain is $1 on some assets. Likewise, if you sold at a loss then you must show a loss on all assets, even if that loss is $1 on some assets.

Basically, when working through an asset you select the option for "I stopped using this asset in 2021" and go from there. Note that you MUST do this for EACH AND EVERY asset listed.

When you finish working through everything listed in the assets section, if you ever at any time you owned this rental you claimed vehicle expenses, then you must also work through the vehicle section and show the disposition of the vehicle. Most likely, your vehicle disposition will be "removed for personal use", as I seriously doubt you sold your vehicle as a part of this rental sale.