Online Etymology Dictionary tells me that the word 'Bully' comes from the Dutch 'Boel' meaning 'brother or lover' originally a 'sweetheart' but the meaning deteriorated from 'sweetheart' to 'fine fellow,' 'blusterer' and finally to 'harasser of the weak', the link between 'lover' and 'ruffian' possibly coming from one form that meant 'protector of a prostitute'
It is also an adjective meaning 'worthy, jolly, admirable', as in 'bully for you', which is a reflection of its earlier, positive meaning. (somebody acting in a brotherly fashion)

The term 'bully' in indeed has less of a connotation with intoxication and more one of being obstinate, stubborn and rowdy, none of which are requisite, but are nonetheless key symptoms of being drunk. The amusing thing is that most of the word's alternate definitions could serve as a reasonable explanation.

it reminds me of the grammatical curiosity 'Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo' where you can construct a meaningful sentence using only homonyms.

In this case, several sailors are granted shore leave and wind up in a public house, The bully, but bullied bullied bully bullied a bully bully's bullied bully, so bullied bully bully's bullies bullied him into alley. (The fine yet harassed and rowdy sailor, in turn harassed some fine gentleman's drunken sweetheart, so other drunken, well meaning pimps girlfriends threw him out into the alley)
---------------------------

But to return to the original topic,

Another thought (SPECULATION ALERT) is that it could be the other way around, with the sailing expression cropping up later on, after the 'bully in the alley' had already come to mean a drunken sailor. After all, a wavering vessel as described before would certainly give the appearance of being steered by a soused helmsman (or that the ship is being stubborn and steering its own course)