To my knowledge, you can make arrangement with the Kegyeleti osztÃ¥ly (Chevra) to keep the grave in order, I can't say about the cost. But you can make arrangement by mail and bank account. On the other hand there are always some people around in the cemetery, who are willing to care for the grave for a certain amount.
I am attaching the email of the Chevra Kadisha and the address and phone of the cemeteries as I found them in the MAZSIHISZ.hu homepage
Kegyeleti osztÃ¥ly (Chevra)
CÃm: 1087 Budapest, SalgÃ³tarjÃ¥ni utcaÂ 6.
Telefon: (06 30) 325-3750
RÃ¥koskeresztÃºri (Kozma utca)
CÃm: 1108 Budapest, Kozma utca 6.
Telefon: 262-4687, 265-2458