After 1 or 2 missions you will have access to the shop and you can access it after you complete each level. You have to complete the game in order to unlock the final gun which is the Trophy gun. You can only upgrade the rifle, long ranged rifle, shotgun, and the Compound bow. Also it is important to know that you only need to buy the best upgrade available for the gun you don't need the lesser upgrades for this achievement to unlock. So for instance on the bow only buy the best ammo, best sight etc and that will mean you have fully upgraded it. I also don't think you need to bother with the weapon skins for this achievement.
So just buy the best upgrades only, they typically cost $5000. You should be able to get this on your first playthrough. If not you'll get this on your second playthrough while going for Gun Master. After you complete the game for the first time you will get $20,000 to spend so that could help.