To start with you need the closing docs from 2008 only because you need the amount you paid. That is the depreciable basis for the property.
There are different schools of thought on land value. I usually set aside 10% and call that the land value.
The IRS allows you to deduct ordinary and necessary expenses. The fact that this is a co-op doesn't really come into play other than the costs of maintaining your rental are paid to the co-op board instead of an HSA or something. Any expenses that you have to pay in order to maintain the apartment and rent it are deductible. In the rental section there are a number of boxes for the different expenses necessary. If none of them seem correct to you there is also a section to write in any expenses that you have.
Your apartment was 'for rent' the first day it was available for rent in December of 2023. Your plan to start the rental period January 1st is fine but you will lose any expenses from December because the expenses need to be taken in the year they are incurred. If you had a lot of expenses in December it may be worth amending your 2023 return and starting the rental with a loss for that December.
The letter you attached isn't legible I'm afraid. But if it lists values from the co-op then that really doesn't apply to your rental business. If it lists expenses that you will be paying then they are deductible as soon as you pay them.
@KathyF
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