I used to use one of those old push cylinder mowers, but it doesn't really like stones on the path. You might be able to pick one up locally. One allotmenteer on our site uses a modern push mower, but its wheels are on the edge and you can only really do the middle of the path. I paid £100 for my second hand stihl and that was a good investment.
Other options are doing away with grass. Not an option I'd go for because I like the traditional grass paths, but gravel or bark chippings are options.