foreign rental income

I started renting out my parents old house in Switzerland for 2022 and have rental income.

I assume you have ownership in the property. If you do not, then you do not and can not depreciate property you do not own. When entering it into the TTX program, make sure you select the option to signify this is foreign rental real estate. That's because foreign rental real estate gets depreciated over 30 years, whereas a domestic rental is depreciated over 27.5 years.

1. I have to pay income taxes in Switzerland on this rental income.  I assume these taxes will get me a credit or deduction on my US taxes?  Is that correct?

Correct.  Though the credit may or may not be dollar for dollar.

2. I won't get my Swiss taxes until late summer or even fall.  What is the best way to deal with this?  Do I pay on all my rental income by the US tax deadline and then file an amended return?  Thanks for any advice on this.

There's nothing to amend. Foreign taxes paid are deductible in the tax year they are actually paid. It *does* *not* *matter* what year those taxes are for. If you pay Swiss taxes in July of 2023, then when you file your US return for tax year 2023 in 2024, you will claim those foreign taxes on your 2023 tax return.