For other potential responders: the nearest town seems to be Komarno (Komarom on the Hungarian side).
I'm currently researching both Slovakia and Hungary for a trip this year. I don't recognize any of the Slovakian towns as having popped up in my reading, but there are several places on the Hungarian side of the Danube that might be of interest. I know James E has posted positive comments about all of them.
It can be difficult to travel between small towns in different countries, but I suspect there may be some public transportation between Komarno and Komarom. Perhaps Rome2Rio.com will be of assistance if any of these places sound interesting. (Forgive the many missing diacritical marks.)
I'm primarily listing specific sights that are art-related. There are of course old churches and other points of interest.
- Baroque town center
- Margit Kovacs Ceramic Collection (Apaca utca 1), Radnai Collection, Kiraly utca 17--20th century Hungarian art. Haven't yet checked opening hours/days.
- Outside town is the Pannonhalma Abbey. Haven't yet checked opening hours/days.
Esztergom: One of Hungary's oldest towns.
- Museum of Christian Art (Kereszteny Muzeum), Mindszenty hercegprimas ter 2, has a really good collection, including paintings. It appears to be closed on Mondays and Tuesdays; verify.
Szentendre: Artsy town with enough tourist traffic to support rather a lot of galleries, etc.
- Ferenczy Museum, Fo ter 6: Works by family of artists Open Tue-Sun 10:00-18:00.
- Jeno Barcay Museum, 10 Dumtsa Jeno Street: Works of a single contemporary artist. Open 10:00-18:00 Tue-Sun.
- Ur Eleonora Batik Galeria, Dumtsa Jeno utca 15: Batik paintings on silk. Operating days and hours unclear.
- Kovacs Margit Museum (ceramacist), 1 Vastagh Gyorgy St. (1990): Daily 10:00-18:00.
- Amos-Anna Museum, 10 Bogdanyi Street: showing works of a married couple (surrealist and folk artist). Open Tue-Sun 10:00-18:00.
- I found a 1990 reference to a Museum of Folk Art of Pest County, Hild Stairway, on side of Templom Hill, climbing from Rakoczi St. to Templom Sq. Have no idea whether it still exists.
- Hungarian Open Air Museum (Skanzen / Szabadteri Neprajzi Muzeum), Sztaravodai utca: Rural architecture with 312 buildings from ten 18th century settlements. Some workshops (handicrafts?). Open Tue-Sun 10:00-17:00. I'd expect the workshops to me mostly active on weekends.
- MikroArt - MikroCsodak Muzeuma, Fo ter 18-19: Microscopic artworks. Allow about 30 minutes. Open daily 10:00-18:00.
- Peter-Pal Galeria, 1 Peter-Pal Street. Commercial craft gallery.
- Metszet Galeria, 14 Main Square. Commerical gallery, reportedly selling antique etchings, etc., as of 1990. Website is not illuminating.
Budapest could keep you busy for a full week, but before heading there to hit specific museums, check their official websites, because quite a number of them are totally or partially closed for refurbishment or relocation at the moment.