To answer your specific second question - no, there isn't any problem with changing one wheel first, except possibly with aesthetics, like if one has a deep rim and the other not, it'll look funny to some people. But no problem functionally.

That said, I'll follow others and suggest spending the money differently. If you're happy with the pedals and shoes you're going to use with the bike, I'd consider spending the money on some nice tires and brake pads, since you're going to need to buy those eventually anyway (assuming you're putting a decent amount of miles on this $5k bike, which I hope you are). Add in some clothing items and you can easily hit $250 before you know it.