From the Airport(Ferihegy) to the Eötvös University Lágymányos campus (ELTE TTK):

Simple: Take an "Airport Minibus" at the Arrivals help desk to the campus (Address: Budapest XI., Pázmány P. sétány 1/A). Costs about 2100 Ft (8 Euros)

Complicated: Take the Reptér-busz (Airport-bus) from Ferihegy 2 Airport to Kőbánya-Kispest, MÁV-állomás (or the regular city bus 93 from Ferihegy 1 Airport to Kőbánya-Kispest, MÁV-állomás). Both cost about 140 Ft (0.5 Euros). Buy 3 tickets from the driver, since you will need it during your long journey. Go until the last/end station Kőbánya-Kispest (about 26 minutes). This is the end station of the metro line M3 (blue). Take M3 and go until "Ferenc Körút". Costs another ticket. Go up to the surface and take a yellow tram no. 4 or 6 . You have to choose the Buda/Danube direction ("Boráros tér felé"). Costs yet another ticket. Immediately after crossing the Danube ("Duna") trough the Petofi Bridge (Petőfi Híd) you have to take off. The stop is called "Petőfi Híd Budai Hídfő". On your left (south) you can see a new building with an observatory on the top. That's where you wanna go.

From the Déli and Keleti Railway Stations:

Take metro line M2 (red) to "Blaha Lujza tér" station. Costs one ticket (140 Ft, 0.5 Euro). Go up to the surface and take a yellow tram no. 4 or 6 . You have to choose the Buda/Danube direction ("Boráros tér felé"). Costs yet another ticket. Immediately after crossing the Danube ("Duna") trough the Petofi Bridge (Petőfi Hid) you have to take off. The stop is called "Petőfi Híd Budai Hídfő". On your left (south) you can see a new building with an observatory on the top. That's where you wanna go.

From the Nyugati Railway Station:

Take metro line M3 (blue) to "Ferenc Körút" station. Costs one ticket (140 Ft, 0.5 Euro). Go up to the surface and take a yellow tram no. 4 or 6 . You have to choose the Buda/Danube direction ("Boráros tér felé"). Costs yet another ticket. Immediately after crossing the Danube ("Duna") trough the Petofi Bridge (Petőfi Híd) you have to take off. The stop is called "Petőfi Híd Budai Hídfő". On your left (south) you can see a new building with an observatory on the top. That's where you wanna go.

By car

While you reach Budapest from any direction the bridges crossing the Danube ("Duna") are well indicated. So, follow the signs directing you toward Petofi Bridge ("Petőfi Híd"). You have to go right until the bridge. Immediately at the Buda-side of the bridge - where you can already see a new building - there is a possibility to go right and leave the bridge. In 1-200 meters you will find the parking area of the Campus and you can leave your car there.