Forget the Holiday Inn - you'll be (literally) next to the main westward 4 lane motorway (M1/M7), 40-50 minutes from anything of interest. You can get nice downtown hotels for cheap, even the excellent Danube-side ones (IC, Marriott, Sofitel).

A 7 days travel card is 4950 Ft (16-17€) with unlimited travel within the city limits. There are some major reconstruction going on on some tram lines, but buses run on those stretches.

Mass transit is good and reliable in the city. Getting to the airport involves taking the metro and at the terminal changing to a bus (#200E, afaik) - a bit long, but doable and included in your pass. The cab drive to the Buda side is ca. 9000Ft (30€) to the Pest side less, to your planned HI more (but you have already forgotten that hotel, haven't you? )