Since you did not use the house as a primary residence you will select only "first year rented".

If a house was used as a primary residence and then converted to a rental during the year you would select both options.    

Thanks.

How do I handle the depreciation in the turbo tax . For example: Bough 60 year old house in 2005 for ( let us say) 100000$  and started renting it starting Jan 2020.

Should I depreciate it starting Jan 2020 ?

Yes, the depreciation would start when you placed the rental home in service (started to offer it for rent).

 The starting point for basis for depreciation is the lower of (1) the adjusted basis on the date of conversion, or (2) the property’s fair market value (FMV) at the time of conversion. 

Please see Publication 527 (2020), Residential Rental Property Cost Basis for additional information.