Am I correct in assuming that the verb attraversare conjugates with the subject & the verb?
In this case "attraverso" means through, so it's not a verb. As a verb "io attraverso" requires a direct object.
Is it correct to say "attraversa la tazza"?
Only if you mean "crosses the cup"... "attraversa" is the conjugation of the 3rd person singular of "attraversare". However, this is clearly not the case. See f.formica's reply.