Rustique is not necessarily peasant. Here, it probably means either crude, ill-mannered or plain, as per
I. A. 2. b)
β) Péjoratif. Rude, grossier, sans savoir-vivre.
γ) Par extension. Qui a la simplicité sans façon et un peu fruste que l’on prête aux gens de la campagne.TLFi
either way, though rustique is not precisely flattering in most contexts, I wouldn’t use it to mean “not good enough” by itself. For instance, here, the self-deprecating implication is conveyed by qu’ rather than rustique.
Yes, rustique can have this deprecating connotation. It can also have a positive connotation in other contexts. For example, charme rustique evokes a nostalgy for the (imagined) pleasant simplicity of the country life of yesterday, the good old days. In your case, un rustique is a peasant, unwashed or at least uncouth and uneducated, ill-mannered, not suited to the high-brow gentrified city life. The negation “je ne suis qu’” makes it apparent that the qualifier is negative (it would come out as probably negative even with just “je suis un rustique”, but the use of ne que makes it unambiguous). I think you got the undertone exactly right.