Moss is an opportunist. If conditions are right; shade, dampness and bare soil then you will get moss. Moss is NOT competing with the grass at all. Grass grown in shade and dampness will die out leaving bare soil and THEN the moss takes over.
If the landlord is amenable, you could get a sod cutter and cut out the moss and grass and dead roots of the grass, dig down a bit more, make a new plant bed with the debris (unless there is the plastic of purchased sod left behind get rid of that part), lay landscape fabric down (this is NOT for weeds and was NEVER meant for weeds) and install the finest gravel you can find. 5/8 minus is a bit large and hard to walk upon. 3/8 minus (you want the fines) is great. Before dumping the gravel on top of the landscape fabric, install 2X4's pressure treated between the gravel and the lawn in the sun. Score with a skill saw for curves and secure with stakes SCREWED into the 2X4 (don't use nails with wood, just screws). Any other type of edging that is thinner is worthless. Trex dimentional lumber that is made from plastic bottles in ONLY dove gray to match the gravel is a good choice as well. More cost initially but it flexs for curves. Careful to do this in warm weather and slowly. In cold weather you could use a propane torch to soften and bend for curves. One also has to drill a hole before securing to the stakes with screws.
Install and compact the gravel. The gravel should be as deep as the top of the 2X4's. Ideally 4". Makes a great informal seating area or patio!
Depends on how much you live out of doors and how you feel about the outdoor room (s) of your rental. I always left a rental with a beautiful landscape. My landlords sometimes paid for the materials and I added the labor. Otherwise, I just did it because I asked and I could and I had lots of materials and plants I dragged home from work. Landscape construction...I got my deposits back though!
If you find moss in the rest of your lawn it is because you are maintaining that lawn incorrectly. I'd just rake it out and reseed. Make sure to read all the stuff on this site on proper lawn care. Don't trust any material from a fertilizer company like Scott's or Ortho...they WANT you to screw up and buy their chemicals to fix the problems us humans created. Because of improper information.
Don't use sulfur or 'moss-kill'. Lawns like a more alkaline soil, sulfur brings the pH down into the acid realm. Then you have to test, lime, test, possibly relime until the pH is around 6.5 to 7.0. If you keep your lawn mowed no shorter than 3", once per week minimum mowing, fertilize with organic slow release fertilizer 3 to 4 times per season, water deeply and allow to dry out (when you walk on the grass the leaves stay down and your foot prints left) BEFORE you water deeply again, get together with your neighbors once a year and aerate leaving the plugs or cores right where they land...You will not have moss in healthy, vigorous grass. A healthy crop of grass will easily compete and win with all the other lawn weeds. And taking care of the yard is just a super way to get exercise, whether it is your home or a rental, yard work is a wonderful part of life.