It all depends upon the quality of the culture that was frozen away. If you are confident that the frozen culture was 100% correct then you can just pick a colony and proceed. But if you have doubts then it is worthwhile to sequence the plasmid before you spend weeks or months doing experiments with it.

In research there is no place to save costs at the wrong step. Always be aware, that there is no 100% in accuracy in any experiment involving living organisms. So, if you pick one single clone from a plate after storage in glycerol, it is by no means safe to assume a correct clone. There is always the probability of mutations. Therefore, you should sequence any DNA after growing bacteria, at least your gene of interest, ideally the whole plasmid, because the mutations could appear anywhere and after generations of clones you might find unexpected behaviour because some parts could be deleted or regulatory elements mutated etc. To save costs, prepare a large batch of your DNA, so you don't have to repeat the sequencing.