Buda

As you may already know, Budapest is actually two cities: Buda and Pest.  Buda is on the western (hilly) side of Budapest and is separated from Pest by the Danube River.

Nestled atop the hill overlooking the Danube, Castle Hill is the prominent tourist attraction in Buda, where you can find a number of lookouts, museums, and a palace. 
As you cross the majestic Chain Bridge into Buda, you are greeted by a traffic circle and the option of traversing through a tunnel through Castle Hill. Unless you are travelling by vehicle, I would not suggest taking the tunnel route as ventilation isn't ideal and you'lll be inhaling a lot of exhaust fumes. What I do recommend though is hopping over the barbed wire fence and taking in the cityscape from the ledge above the tunnel.

Fishermen's Bastion is the most impressive lookout structure at Castle Hill.  

I'm not sure the admission fee to the second level is worth it because it appears as though the view would basically be the same as from the free, lower level. Nevertheless, there was a steady flow of tourists heading up there. In my opinion, sitting back and enjoying a drink in the Bastion's cafe would be a wiser bet.

Mátyás Church is another prominent feature on the Hill. I didn't spend too much time here as by this point, my appreciation of church architecture was becoming blurred by this point due to overexposure, but here's a sneak peak inside in case you're curious:

I also did not visit any museums or galleries in Buda or Pest due to time constraints and quite frankly, the lack of interest as I preferred enjoying the very pleasant and warm, autumn sunshine, and hanging out in the backyard of the Office of the President of the Republic of Hungary but let's not get into that...

I'm not sure why but perhaps it was the walled city or the narrow, cobble streets, but Castle Hill reminded me a lot of Quebec City.

Other unique attractions in Buda include the Cave Church, which not surpisingly is a church in a cave,

and the Gellert Baths, one of the famous thermal baths in Budapest, despite the apparent spelling mistake...

Now one would think that considering the number of tourists visiting this area of Buda, there would be a sense of pride and obligation to keep at least this section of the city, clean and very presentable to outsiders, but there were numerous instances I came across in which this does not seem to be the case...

  

I was also surprised to see a major historical Castle Hill structure in plain sight to tourists boarded up and simply rotting away

That being said, I understand that restoration costs money but I would have thought there would be a little more civic pride in at least cleaning up historical monuments to entice tourists to stay in Budapest even longer to support the financial costs of maintaining these impressive monuments.  Nevertheless, this apparent ignorance to the city's historical past, adds to the unique character of the Budapest.