The night-train (D407 'Chopin') leaves Warszawa Centralna at 20:20 and arrives Budapest Keleti at 08:35. If you want a full day in Warsaw, followed by a full day in Budapest, with a reasonable night's sleep in between*, I would say this is almost certainly a better option than flying.

According to vagonweb.cz, the sleeper carriages on this train alternate between those operated by Wars (Polish railways) and Utasellátó (Hungarian railways). Looking at the photos, they both appear to have been originally built in East Germany in the 1980s, although I think the Wars ones have been upgraded since then. However, although they're a bit old, I actually quite like these carriages - they have the basics (washbasin; electric socket (which may or may not work), etc), but are much more spacious than more modern sleeper compartments, and they have windows you can open (better for photography during the day time, etc). They don't have showers.

The train also has a couchette carriage (presumably also from a similar era), but it's probably worth paying the relatively modest extra amount to get a sleeper berth (in a shared compartment - single compartments are nearly always significantly more expensive...)

Unfortunately, PKP (Polish Railways) don't appear to sell tickets for this train online. So, if you want to get your ticket (and, more importantly, sleeper reservation) before you arrive, you'll probably have to use a travel agent. Polrail seem to have a good reputation, and can sell tickets for this train. However, they charge 567Zl (~$180) for berth in a 4-berth couchette compartment, up to 633Zl (~$201) for a berth in a 2-berth sleeper compartment, or 1,023Zl (~$325) for a berth in single sleeper compartment.

These prices are all somewhat higher than I would have expected (apparently MAV offer tickets in the other direction, starting at €49 for a berth in a 2-berth sleeper; and Warsaw-Prague in a 2-berth sleeper was ~€70 a couple of years ago (purchased in Czech, not Poland, though)). It does seem that Warsaw-Budapest is a more expensive route than Warsaw-Prague, and that tickets starting in Poland are more expensive than tickets in the other direction. However, if you will be in Poland for a while before you travel to Budapest, you should be able to buy your ticket directly at a station in Poland, which should be somewhat cheaper than the travel agent price. (Although of course, if you try to book at short notice in peak season, you may find that there's no availability left...)

More information is, of course, available on Seat61...

* Some people find it difficult to sleep on night-trains, even in a proper sleeper berth. I wouldn't recommend travelling overnight in a seat, but I can sleep adequately in a couchette, as well as a sleeper. And heading out to an airport for an early morning check-in tends not to allow much sleep, either... YMMV.