I have a little less than an acre. Its in terraces 3-4m wide. There are several and the ground is quite steep. I have an access path between them. The steepest point of the path is probably a 2m drop in 4-5m horizontal
Can ride on mowers cope with that?
A strimmer seems like a huge amount of work?
A petrol push mower? How much work is an acre with one? How long would it take?
String trimmer is no good for actual mowing. It would be perfect for all the edging along the walls though, if you want to get it right to the edge.
Ride on mowers will have issues with steep uphills especially if the mower deck is engaged. I would ride up and then mow on the downhill if the power from the mower is limited. You would benefit greatly from having traction ballast in the tires (put in tubes, then fill them with calcium chloride solution).
We have about two mowing-required acres with some more gentle slopes. With an old 38 inch John Deere with 12 hp Kawasaki engine, it took probably 3 hours and you'd have to be careful to overlap swaths so as not to overload the mower. With a 54 inch Kubota tractor and a 22 hp diesel, it took about 1:45 and you only need to overlap so as to ensure you don't miss swaths. Biggest hassle was that the front-end loader impedes vision and maneuverability around trees. Then we got a 48 inch Kubota zero turn mower. Got the gasoline engine since the diesel one was nearly double the price (and used ones were not coming up for sale in our area). With it we can mow everything in about 1:30, and the biggest challenge is to not accidentally run over small shrubs and trees... the independent rear-wheel steering intrinsically gives you good traction on hills but traversing along a side slope can result in a crabbing action since the front wheels just caster.
If you can use a tractor with a loader I'd consider that. The newer ones (from Kubota at least) have an easily detachable loader including quick connects for the hydraulics. The four-wheel drive is amazing. If you only need it to mow and you have the budget I'd look for a zero turn radius mower. It will cope with your switchback terraces like no other.