Bummer appears in 1855 (Portland Oregonian) and bum in the 1860s (Gold Hill NV News, 1864) in American print (in the sense in the song). Probably older and both probably equal in age.
Bummer in print in England in the 1860s (Pall Mall Gazette) in the same sense. Country of origin??
Hobo is more recent, 1880s in U. S. print, Ellenburgh, Washington, Capital). A western U. S. term?
Vagrant, the term beloved in legal statutes, goes back to ME, 15th c. in print.
Bummers Shore may come from one of the song versions which includes the words, "bummers, sure,..." Then again, could be just some singer waxing poetic.

Bum as buttocks goes back to the 14th c. at least, from ME bom.