You can walk into the city center... it's pretty quiet that time of night, but beautiful (nice neo-gothic sky scrapers, some interesting modern architecture), and safe.
For a wonderful jaunt, catch the (underground) street car, from the city center (a few blocks from the station) a couple of stops across the river to Station Square. This is a little mall built around the old Pittsburgh and Lake Erie station, a gorgeous mini-Grand Central, now home to a fancy restaurant and oyster bar. If you can afford the restaurant, it's worth it, but if not, you can still look in, and eat at one of the mall-type chains. No more than $10 in a taxi if you don't want to play with the trolley, but of course you feel less of a connection to the city.
Don't know about coffee in the morning, but on a weekday the city rises early.