If the decor is fairly modern you could get away with furnishing it from IKEA, they have 2 stores in Budapest. There is also a French IKEA-style store in the 13th that's reasonable, can't remember the name of it, maybe Budapest can tell us.
If you walk around Ikea and make a list of the codes of all the furniture you need, and add up the prices, it will give you an idea of what it will cost. You can get the white goods and telly etc in either Tesco or Media Markt; if you are in the IKEA at Ors Vezer Tere (last stop on red line) there is a Tesco and a Media Markt just across the road, so you can do the lot in one place. I reckon you could furnish a 2-bed apartment in this way for about 6K at a pinch. Only thing is that if the apartment has a heavy traditional decor then the IKEA stuff won't look great in it.
The small stuff, cups and plates, cutlery etc you could pick up in Tesco, or just let the management company do it -- life is too short to shop for plates.
If you have made a list from IKEA and of the white goods, give it to a management company (again, maybe Budapest can help you) and ask them to buy and instal everything for you. Typically they charge 10%, and they also usually get a small discount from IKEA but not from Tesco or Media Markt, so they should be happy enough to do it for 10%. Then ask them to rent it for you and manage the rental. Remember that 10% or 15% of your rental doesn't amount to a hill of beans, so be reasonable with them and don't expect them to work full-time for you for this money.
Open a bank account when you are in Hungary (you can't do it from here)and put enough in it to keep it open while you are waiting for rentals to arrive -- if you just put in a nominal amount the monthly charges will eat it up and the account will close, requiring another trip to Hungary to open another account.
All this may seem like a lot of trouble, but its really just a days work, and you need to make the most of the situation and get some cash flow from the investment. Overall, I wouldn't be too worried, you could have bought a lot worse than the middle of Budapest, and I would reckon that it will do ok for you in time despite your bad start.