Brussels has an extensive public transport network. The STIB-MIVB website has more information.
There are a few different bike share options:
Brussels airport is located just north of Brussels, in Zaventem. It's well connected to Brussels by rail.
The SNCB-NMBS Journey Planner is a good place to plan this trip. Tickets can be bought in the airport.
There are direct trains from the airport to Etterbeek station, which is right next to the VUB campus. Look for IC trains towards Namen/Namur and Dinant, or local ('S') trains for Brussel-Schuman and Charleroi. Either option takes a bit less than half an hour. Many non-direction options exist as well, and there may be better options for other destinations (e.g. hotels).
The airport known as Brussels South Charleroi Airport is located near Charleroi, around 40km south of Brussels.
The easiest way to get to Brussels from Charleroi airport is a shuttle bus. More information can be found on the Charleroi Airport website.
There are multiple long-distance train options to get to Brussels. The best rail company to book through will depend on your starting point. Eurostar and DB are likely options.
Driving to or in Brussels is not recommended.
If you do, remember that Brussels has a low-emissions zone and foreign cars may need to register. See the LEZ website for more information.