You can easily spend more than 4 days in Budapest.
Personally, 3 days in Eger was enough for me. Eger is really best for a week-end visit - Friday and Saturday nights (even in October) are fun and busy in the valley of the wine cellars! One visit in late October, the wine shop in the little plaza in from of the Senator Haz hotel provided wool blankets to drape over my lap while I sat outside sipping wine... it's a very pleasant way to wile away an hour or two in the afternoon!
From Budapest, I would highly recommend joining a tour to Godollo and "Sissy's Palace" - some of these tours also include what is best described as a Hungarian cowboy show. It's an excursion that I enjoyed very much. Another great day tour from Budapest is to the Danube bend.
I also joined a Jewish Heritage walking tour that included the Dohany Great Synagogue, the Holocaust Memorial and access to a small museum. Although not Jewish, I found the tour very enlightening and moving.
If you like wine, the Agricultural Museum (near Heroes Square) had a neat small wine museum in the basement.
I also had fun finding a few of the street sculptures by the artist Imre Varga, including the famous umbrella statues in Obuda.
The indoor food market was the best place for lunch (and to pick up small souvenirs, like paprika).
The thermal baths are a big reason why I like Budapest so much! Usually around 4-4:30pm, everyday I headed to one of the thermal baths. I liked the Lukács bath best and I'm really looking forward to trying the newly opened Veli Bej (Császár Baths) when I go back next May. The Veli Bej should be the one which I think should top anyone's list to try, if time only allows one. Both the Lukács and Veli Bej are on hospital grounds and have therapeutic waters - perfect for sightseeing- and travel-sore bodies! ;-)
Getting there is relatively easy by tram or bus, with a short walk from Margit hid. By taxi, you must use the word "furdo" after whichever bath's name that you picked to go to.
When you buy your entrance ticket, this is also the time when you can book a massage - which I also recommend at either of the two baths mentioned above. Before the big reno (when I was there), Lukács had rare tourists and was very institutional. During one of my earliest visits, I had a communist era-style massage by a great lady (who had obviously been trained by 'Olga'/"I will pound you until you feel better!") - LOL, but it was good!
I am hoping that the renovated Rac baths will be opened by May (but likely not, since that project seems to be a bit jinxed). Tuesday is Ladies' day at Rudas and "mixed" days are on Friday and Saturday - I found this one to be the most atmospheric of the Turkish baths. I found the Gellért, Széchenyi, and Király just OK; with the first two quite touristy, IMO. The mineral thermal waters share health benefits on the Buda side. Széchenyi's waters, on the Pest side, had to be drilled for deep and doesn't have the same minerals. This has to do with the Buda side's karst geology.
BTW - I did get good value out of the Budapest Card on my first visit.