Does TurboTax automatically file Form NYC-202S (Unincorporated Business Tax - UBT) on behalf of us, or do we need to print that form out separately and file it ourselves?

You need to print and mail it; e-file is not supported.

See https://care-cdn.prodsupportsite.a.intuit.com/forms-availability/turbotax_ny_online_individual.html

HOW DO I GET INTO THE NYC-202S TO FINISHED MY TAXES

Form 202S is not supported for efile.  It would need to be printed & mailed in.  See the link below for form availability:

• Form Availability