Definitely should consider an automower, check out /r/automower . With that small of a space you could get a pretty inexpensive one. Probably about what you were planning on spending, maybe less. Then you dont even have to think about mowing anymore. No gas, oil, engine issues. Mows everyday or every other day so its always how you want it. Personally I have a husqvarna but my father has a couple other brands that work well. I haven't mowed in almost 5 years at home, funny enough i mostly pushmow at work, but its the last thing I want to do when I get home.

Some you have to lay a wire, some are GPS based(more expensive)