Hong Kong Museum of History
Admission is only HK$10 (that's really cheap with the exchange rate!) and the museum is open 10:00-18:00, 10:00-19:00 on Saturdays, Sundays, and holidays; closed Tuesdays and a couple major holidays).
The museum is accessed in a few different ways. The easiest and closest MTR station is probably Hung Hom, using exit D1. East Tsim Sha Tsui (P2) and Tsim Sha Tsui stations (B2) are also nearby. Kowloon isn't too spread out so walking isn't out of the question. The museum's website is here and has some maps to help you get an idea of the museum's location and where the entrance is on the building.