How to tax a primary residence converted to rental last September?

" I guess the 2021 real estate tax and mortgage interest for our old residence need to be split  to reflect  the primary-residence period and the post-primary-residence period."

Yes, this is what you do. Prorate the mortgage interest and property taxes by date. Enter the personal portion on Schedule A and the rental portion on Schedule E. You will have to do the math yourself.

You'll enter these expenses in the Rental Expenses section under Other Common Expenses. (Click the screenshots below for reference.)

To get to the rental expenses section:

• Click My Account (top right of your screen).
• Select Tools.
• In the pop-up window, select Topic Search.
• In the search bar, type rental.
• Scroll down in the results box, highlight rental (schedule e), then click GO.
• You'll land on Income from Rentals or Royalty Property You Own. Click Yes.

Assuming this is your fist time as a landlord, the below information will help to clarify some things for you that the program does not clarify very well. Note that absolute perfection in the first year of renting is not an option. It's a must. Even the tiniest of mistakes will grow exponentially over time. Then when realized years down the road, the cost of fixing it will be high.

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.

@Mishonio I think it is not necessary to split real estate tax. The whole amount should go to the rental property section, as TT highlights at the bottom. 

@Mishonio  scratch what I said, we SHOULD prorate the tax as well. Please see the reply from @mglauner and @DianeW777 below