First, take public transit. If not, I'd recommend just getting a car at the airport. It shouldn't be more than $100 for 36 hours. You'll spend $60 on taxis to/from the airport.
If you decide to take Metrolink/Metrobus, there are two stops near the Hilton and two stops near Wash U. A day Metrolink pass will run you $7.50 per person, and is good til midnight the day you buy it. You could do that for the day (and metro to the hotel, then to Wash U).
For Pappy's, you can either take the #99 bus to the #97 bus and a short walk, or walk up Washington Avenue (with lots of restaurants/bars/etc) and then take the #97 bus to near the restaurant.
If you don't want to go all the way there for food (though it is pretty dang awesome), I'd check out Terrace View (at City Garden) or any number of places in the U-City "Loop" near WUStL campus.
Other places you may want to check out while in the city:
City Garden
Forest Park (along with the Art Museum, History Museum, Science Center)
Missouri Botanical Gardens
City Museum
Anheuser Busch Brewery
My wife is in a doctoral program at WUSM - it's a great school both for undergrad and grad school - plus living in St. Louis is reasonable and fun.