I'm still on my first playthrough of Bastion.
If I want the Man-at-arms achievement, can I finish it up at leisure in New Game+ or do I have to upgrade every weapon before I go in for the final mission in this playthrough?
The upgrades you have done will carry over to your new Forge in your New Game+, so you will be able to collect items and such to upgrade your weapons. You can even upgrade weapons you don't have yet in your NG+ playthrough (I was able to upgrade the Dueling Pistols, for example, at the beginning of the game).
So yes, you can complete the game and still have the ability to finish up that achievement in a NG+.