I've got no problem with Josh having been there the whole week. He's a pretty open character, so anyone who's been in the cell with him and either asked about his past or listened when others did would pretty much know all the details in his character background (Which you can see here). The only notable exception is his family name; I figure his pseudonym is designed to protect his family from recriminations for his acts, so he's like to keep that close to the chest until he really trusts someone (you'll notice he used the stage name when introducing himself to Orb, for example).
Alternately, the background sets up "Josh o' the Mornin'" as a minor local celebrity, so the name itself might ring some bells for locals even without him having said much else. In that case, they probably wouldn't know his family background, but just that he's been known to do a lot of satire/protest performances and has developed some good will with some of Trenchtown's working class because of it (of course, there are probably also members of the working class that resent his "making a name" off their suffering, so the rep could go either way).
jason
Alternately, the background sets up "Josh o' the Mornin'" as a minor local celebrity, so the name itself might ring some bells for locals even without him having said much else. In that case, they probably wouldn't know his family background, but just that he's been known to do a lot of satire/protest performances and has developed some good will with some of Trenchtown's working class because of it (of course, there are probably also members of the working class that resent his "making a name" off their suffering, so the rep could go either way).
jason