>the eye (Ferris wheel)-does anyone know if it is open in April?,
There is no mention of specific opening or closing dates on the official site: http://oriaskerek.com/visitor-information/
This type of wheel is quite common across Europe. Unless weather conditions are really poor (e.g. very high winds) they operate throughout the period they are in a location (might be years, might be months, might just be for one special event). But, to be honest, it's not something I'd bother taking such small children on. They're not as amazed at being high up as older children and are basically just cooped up in a 'gondola' for the length of the trip.
> is there anything else the kids would really enjoy?
What children of the ages you mention really
enjoy is lots of space to run around, jump, climb and explore. Being in a 'foreign' place is not really a big deal for them. So I suggest you take advantage of the parks in central Budapest
as well as visiting Margaret Island
, which is traffic-free and has plenty of space (and a petting zoo).
If you're going to Budapest Zoo & Botanical Garden
anyway is makes sense to visit Holnemvolt Var
because your zoo tickets cover the entrance...and if you're not impressed you can just leave?
I think the puppet shows are in Hungarian but the stories are simple, the presentation very skilful and it's easy to follow. Small children are anyway not as 'thrown' by lack of language as adults are. The official site (in English) tells you what's on: https://budapestbabszinhaz.hu/en/performances/whats-on/
is an interactive science museum which has its own 'Tiny Explorers' route and imo is definitely worth visiting:https://www.csopa.hu/en/
If you're there at the right time, the Children's Railway would make a good trip and includes stops where running-around is an option. Official site in English:https://www.gyermekvasut.hu/english_eng/
Lots more ideas for kids here:http://visitbudapest.travel/guide/budapest-with-kids/