To upgrade a weapon go to any Friendly Fire and open the 'Buy/Upgrade Weapons' menu.
You have two choices for this, you can either find the secret 'Tiny Pistol' weapon at the South 'Lets Pretend' clothing store and upgrade it since it has only one slot to upgrade (Recharge Rate) or you can buy the Baseball Bat for $100 which costs $42,600 to full upgrade and is probably the cheapest method.
If you wish you can save just before doing this, upgrade your weapon, quit and then reload to keep your money for other things.
Here's a video by TheReviewDemon showing you where the secret 'Tiny Pistol' is.