What is the Currency in (Budapest) Hungary?
No, it is still not the euro, it is the Hungarian Forint (HUF or Ft) (say: four-int).
The Hungarian Forint has no change, Hungarians big banknotes and small coins forint alike (small value coins used to be Filler but not any more).
Can I pay by Euro in Budapest, Hungary?
Euro is accepted at several places in Budapest, but you can never know which places (it is absolutely random). So your safest bet would be to find an Exchange Bureau (abundant in the city center, in plazas and in the shopping street Vaci utca) and change some of your currency into Hungarian Forints.
Even better, you can safely rely on your credit or debit card as most major credit/debit cards are accepted anywhere in Budapest (from stores to restaurants), like Visa, MasterCard, American Express etc.
And, unlike in Japan, you cannot pay with your cell phone yet. :)