I would just tell the landlord. Then go from there. What would you do if your a/c broke? Same thing. Call the landlord.
Usually, a sprinkler head isn't a big deal to repair. I personally have never done it. We had one break (well actually, it got dirt or something in it so it didn't go back down fully, and then hubby ran over the top with the lawn mower, lol). Anyway, the guy who installed our sprinkler system, and has done other subsequent landscaping for us, was scheduled to be out in our neighborhood doing something else, so I just called and asked if when he was out here, he could stop over and repair it. He sent over one of his guys and fixed it. No biggie.
Not even big enough for him to send me a bill for it. :)
Seriously, the parts for something like that probably cost about $5.