Yes you can probably repair it yourself. You did not say what make your faucet is, but following is some info for Mansfield. What you are describing is a leak around the anti-syphon valve. Hope this helps.
Mansfield puts out two kits, one #630-7500 to repair the anti-syphon valve, for around $3.00 and the other #630-7755 to rebuild the rest of seals and washer for around $7.00. They should be available at most plumbing supply houses, etc.
Many times a quick repair for a leak around the anti-syphon cap, is to pry off the plastic cap with a screw driver, being careful not to break it. Work a little from two sides until it pops off. You can then unscrew, counter clockwise, the bonnet (valve retainer) with a pair of plyers. Clean any crud from under the bonnet, which acts like a washer seat. You will find a plunger assembly, clean and reverse the washer in the assembly. Re-assemble the parts you just removed. Tighten the bonnet firmly. Snap the cap back on. With a little of luck you will get many years of additional service before needing a repair kit.
If you just have a drip from the faucet, your best bet is to obtain a rebuild kit and replace the washer and seals.