You might be able to get the Al Seef Heritage Hotel (Curio) for around that. I just did a stay there and enjoyed it. It's in the redeveloped Al Seef souq area in Bur Dubai, the old city, which has a bunch of old buildings (or made to look old) in traditional architecture. The hotel is spread around ten buildings in the souq. I had a nice room in an old style, including a balcony over the creek. Service and breakfast (included for Diamonds) was minimal but the room was comfortable enough.

The shops and restaurants are touristy and not very interesting in my opinion, but the environment is decent along Dubai Creek. There are water taxi stops along the creek-front. You can use the pool and bar at the Hampton Inn and Canopy close by.