At WAW, stay airside—compact terminals with good dining are safer than navigating city transport.
A 3-hour layover is tight but manageable if immigration is fast. Take the SKM train to Warszawa Centralna (28 minutes, PLN 4.40). Walk to Nowy Swiat, the main pedestrian boulevard, in 10 minutes. Grab coffee at one of the many cafes and walk north toward the Old Town entrance. Head back after 1 hour.
Take the SKM train to Warszawa Centralna and tram 22 or 24 two stops north to the Old Town. The Royal Castle is PLN 30 and takes 1.5 hours. Walk the Market Square and stop at Bar Mleczny Prasowy on Marszalkowska 10 for a traditional milk bar lunch: filling and under PLN 30. Leave by hour 5 to give yourself time back.
Add the POLIN Museum of the History of Polish Jews (PLN 25, 2 to 3 hours) in the Muranow district, 20 minutes north of the Old Town by foot. The building and the permanent exhibition are both exceptional. This works comfortably with a 10-hour layover.