The market value of the property at the time you inherited it does come into play (that is your basis before property improvements were made), but so does the fair market value of the property on the day that it is placed into service as a rental property. The basis of the property for depreciation is the lower of those two values. Then, you can add the cost of the improvements to that basis.
The cost of the improvements cannot be deducted in the year that the work was done because the property was not considered to be a rental property until it was being advertised and made available for rent. Until it is placed into service as a rental, any expenses you incurred in getting the property ready to be rented are personal expenses and are not deductible. This includes transportation, food, and lodging. These kinds of expenses are only deductible if you incur them after the property has been placed into service as a rental and the expenses are related to your maintaining or repairing the property at that time.
Improvements made to the property before it is a rental are allowed to be added to the basis for depreciation because they become a component of the property itself. This would be true even if you never actually rented the property, but instead you sold it. The costs of the improvements would be added to the basis in that case as well.
To learn more about the basis for depreciation for the rental property, take a look at this information from IRS Publication 527.
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