My flight arrives at Budapest at 1:00, and the next flight leaves at the same day at 23:00. I need a hotel to leave my luggage and have some sleep during this 22 hours period. But, usual hotels can be booked only from 14:00 to 10:00 the following day. Is there a way to book a hotel without having to book two nights?
As suggested, AirBNB can be helpful. You can contact several hosts prior to booking their property and explain your special requirements. Based on that you can book accordingly.
There are several other options like sleeping at airport terminal lounges for a few hours and exploring the city while getting a storage locker for your bags. http://www.sleepinginairports.net/ is a good guide for sleeping options in or around any airport.
Also you can book hostels which might be cheaper to book so it may make sense to book two nights if you find a good price and stay in a central area so you save on travel costs while exploring the city.
If you just want to sleep in a bit and go explore, some hotels have later check-out than 10:00; try looking around a bit on booking websites, and you might be able to find 11:00 or 12:00. Others are willing to extend check-out time for a fee; there's no harm in asking, if you find a place where you'd otherwise be happy to stay.
Alternatively, many Airbnb rooms have later or flexible check-out times, especially if there's no guest the following night (be sure to double-check, though, since some have early check-out and are fairly strict about it).
As far as staying in your room through the afternoon (as opposed to later in the morning) without booking a second night, you're probably out of luck, unless you find an abnormally permissive Airbnb host, small B&B, etc. But if you're only in the city for one day, you probably will want to be out in the afternoon, and most hotels will be willing to store your luggage for the day after you check out.
This is probably not a hugely useful answer unless the OP miraculously fits the criteria or can status match. However, for completeness -
A Starwood Preferred Guest Member who reaches 75 Eligible Nights with Starwood Preferred Guest during a calendar year will qualify for the Your24™ benefit through the following subsequent calendar year.
As a qualified member, you can request, any check-in time for an upcoming stay. Requests must be submitted by 3pm, hotel time, two days prior to arrival. If the request is confirmed, your check-out time (day of departure) will be same as your confirmed check-in time
If you arrive at 1:00, you can be at hotel at about 2:00, so you still have about 8 hours until 10:00. Considering some bathroom and preparations, you should still have about 6 hours of sleep, which is, for most of the people, perfectly fine.
As for you luggage, hotels will usually allow you to store your luggage after check-in and take it later. They have no interest in not giving you that option - it costs them almost nothing, and if they refuse, the folks will write some negative opinions, and some people will go somewhere else. From my experience, it's not a problem, but you should ask to be sure.
So, if you don't need to sleep between 10:00 and 22:00, you should be perfectly fine.
Assuming your main concern is saving money, simply book two nights at a cheap hotel. While there are hotels out there who wouldn't mind an unusual check-in time, these are more likely to be on the expensive side. Booking a $200 room with 24/7 check-in doesn't save you anything when compared to getting a $50 room for 2 nights.
The reasoning for this is simple: hotels have strictly established check-in/check-out patterns and violating them costs time, effort and headache. Someone has to pay for that extra effort and that is always you, the customer.
There are some hotels that will allow you to check-in anytime the desk is open. One that I know of is Corvin Center Suites (hotels.com link).
Check in is open from 4PM to 1AM, but if you are arriving after 1AM just inform them in advance:
If you are planning to arrive after 1 AM please contact the property at least 48 hours in advance using the information on the booking confirmation. Airport shuttle service is available on request. Contact the property in advance to make arrangements. Guests must contact this property at least 48 hours prior to arrival to arrange check-in. Failure to do so will result in a surcharge. Please contact the office using the information on the reservation confirmation received after booking.
I was researching them during a trip last year, and remember this to be one of the reason they were shortlisted.
As far as I can see the other answers are not Budapest specific.
There are, however, Budapest specific answers but these often do not have English webpages. For example, http://www.lakas-orakra-budapest.hu/buvohelyarak-budapesten.html is specialized in renting out apartments in the center per the hour, they mention these are used as a meeting spaces for couples but also mention transit passengers 13-24 hours costs 16500 HUF, 53 EUR, that's for two people. That's very reasonable for an apartment, I'd say. This place asks half to be wire transferred first when someone uses them on the first occasion. Transferwise is your friend for that.
This one http://buvohely-kispest.com/arlista.html advertises a place in the XIX district which is quite close to the airport actually and says 24 hours is 13 000 HUF.
This one is also quite close to the airport http://buvohelyterkep.hu/budapest_budapest-xviii-kerulet_gloriett-buvohely_buvohelyterkep-299.html says 24 hours is 14 000 HUF. While the building both for this and the previous one is ... ahem ... let's just say more authentic than you ever wanted to see (giant, drab, prefab concrete buildings from the socialist era -- I guarantee you that most tourists never see one of these!) ... the apartments are said to be air conditioned which makes them bearable (otherwise in the summer they become ovens and the central heating in the winter also makes them way too hot -- I grew up in buildings like these). This site is actually a search engine for this kind of apartments.
Google find a lot of these when I enter búvóhely órákra budapest but I have no direct experience with them and considering their primary usage is pretty dodgy I wouldn't be surprised if they would be dodgy too. And there's the language barrier. But, this is what's on offer. I reached out to some primarily airport focused places, hotels and guest houses (panzió) both and they refuse to budge on fixed check in and out time so I really doubt there's anything else but apartments either via one of these "hidey hole for a few hours" websites or airbnb.
In Asia I use Red Planet. Tune,Red Dragon. For this. You put it in special request before you book on line. You get a yes or no back. Before you book. Days inn will do this also. So check mid range hotels. Many will do this. Many cheap hotels will also do this but you might loose luggage at them. I think I am on Redplanets friend list in Asia so they cut me some slack.