That's in the modern sense though, there was a revolver called a'Bulldog' back then.
The modern sense of the term didn't arise until the mid-20th century, and the firearm configuration it denotes was first patented in 1901 (too late to have informed the Holmes story).
'Bull dog' / 'Bull pup' were supposedly 19th century American slang terms for a pistol one carried for personal protection.
More specifically, the name 'Bull Dog' was given to a relatively compact Webley revolver, manufactured from the 1870's onward, and widely known among military circles beyond the UK (e.g., Custer purportedly carried a pair of Bull Dogs at the Little Big Horn). The Bull Dog was designed to be carried in a pocket.
The name 'Bull Dog' was engraved on the larger caliber (.40 - .45) versions. However, Webley also made smaller caliber versions (.320 - .380) that were not engraved as 'Bull Dog' models. It's entirely conceivable that 'Bull Pup' was contemporary slang for one of these smaller-caliber variants. It's also conceivable that Conan Doyle (never one for exacting details) used 'bull pup' to connote one of the better-known large-caliber Webleys.
As time goes on, I find the pistol theory less and less satisfying. Watson says "I keep a bull pup" in response to Holmes' request to list any characteristics that may prove to be issues in their sharing quarters. "I have a handgun" doesn't strike me as a negative factor Watson would have mentioned in this context, and the use of "keep" is just as appropriate for two other explanations:
(a) The obvious - i.e., Watson was keeping a bulldog pup. This could have been a potential problem in relation to Mrs. Hudson's terrier, so it would have been something worthy of mention in response to Holmes' query.
(b) I've seen claims that "keep a bull pup" was 19th century Anglo-Indian military / colonial slang for "have a short / notable temper." However, I've never seen a specific source confirming this alleged usage. This, too, would fit in the context of listing a potential roommate's down sides.