If you stay in the Gozsdu Udvar I would suggest an inner facing apartment. That place can get loud at night. The location otherwise is very good and close to quite a bit, but the statement:
and close to the chain bridge, parliament, fisherman's bastion, St
Stephens Cathedral and the boat docks for river cruises. A short tram
ride from the Central market hall, the city park, Szechenyi baths, and
I am certain was based on romantic memories of a beautiful city, but is a bit overly enthuastic. The Chain Bridge is a solid 15 minute walk, fishermans bastion a solid 30 minute walk and there is no tram from the udvar to City Park or the Szechennyi Baths; but you can reach both with a 5 minute walk followed by something less than 10 minutes on the M1 metro. It does lead into the old ghetto and there is a lot of night life in the courtyard. And its close to the 47/49 Tram stop which is also convenient.
There are a lot of good hotels well situated and a lot of short term rental apartments well situated. For a week long stay I would go for the apartment for the reasons David stated and a substantial savings in cost. Its nice to come "home" at the end of the day. Air Conditioning is a must, but always ask, because its not the norm. I travel to Budapest 2 or 3 times a year (returning this Friday) and long ago switched to staying in an apartment. For shorter stays it can be a toss up between hotel and apartment.
The place David recommended is on a side street about two blocks from the Opera House and the Opera House stop of the M1 Metro. The M1 goes from City Park and the Bath House at one end of Andrassy ut to within a block of the river at the other end, so the location works well. My favorite wine bar is at the end of the same street. Again, a good location.
For hotels if you go that way, the best all around hotel in town for cost / location / quality I believe to be the K&K Opera which is on Andrassy ut next to the Opera House. If you read my posts you might guess that I like that part of town. Pretty ideal for tourism.
I would not recommend Buda. You will spend all your time going to Pest.
If you find a place you like, PM me the name or the location and I might be able to tell you about the neighborhood.