I like Kiraly utca as well (I always stay very near Kiraly utca), and the Wombat does get good reviews. Most importantly you only have two days. The first day you are going to be tired from not sleeping on the overnight train. You will arrive at a ridiculously early time of the morning and you will stink; or if not, you will by the time you get checked into an accommodation with a shower. So if I am correct in all of that, and if you don’t have too much luggage, you might consider heading from the train station straight to the Széchenyi Bath House. It is one subway connection away (take the M2 to the M1 at the Deák Ferenc tér Metro Station and then the M1 goes straight to the bath house). It will cost you about $1.30 for the metro ride (one ticket, the transfer is free). At the bathhouse get a Cabana Room to change and store the luggage. The bath house will cost you about $18.00 (sorry) for an early morning ticket. Enjoy the morning soaking in neo-classical Hungarian opulence.
Now go meet the manager or owner of the apartment you rented. They will generally let you in about 11am. I would suggest as the previous person suggested that you get a place near Kiraly utca; or maybe between Andrassy ut and Kiraly utca. That puts you pretty much in the cheaper part of the center of town and close walking and metro proximity: http://1drv.ms/1Ji1bD4 In June you might find a nice place for about $65.00 a night, and if you are really on the cheap now you can cook too. It’s June so get a place with air-conditioning and a washer/dryer might help as well (it’s part of how we travel so light).
Cultural? Well, in most of Central Europe the cultural season is wrapping up in June. I checked and there was nothing at the Opera or the Operett for your dates. The Palace of Arts had a few things and while it is a very modern venue, it is beautiful. That leaves the tourist venues like the church concerts and the folk dance shows and the river cruise shows. I can’t help you much with those but you can google for them or check here: http://www.budapestbylocals.com/budapest-concerts.html
For what to do; I would head for Buda immediately after checking in. See the old Castle District, the Hospital in the Rock if that sort of thing interest you. Walk there and walk back. Day two I would devote completely to Pest. You can easily spend a full day in Pest for the cost of a few tram rides, and still not see even a third of the “good stuff”.
(Option B): Get a hotel room. A good moderately priced choice is the K&K Opera at about $125 a night last time I checked (or the Womabat). Leave your bags at the front desk until the 2pm check in time and start seeing the town. I guess if you stayed at the K&K or the Wombat I would still suggest you take your tired stinking body up Andrassy ut towards the bath house. The mansions along Andrassy ut, the Bath House (to use or just to see the architecture), Hero’s Square are all worth the trip up Andrassy ut. Do remember that most if not all of the museums are closed on Monday.