Google "ground cover for shade" and your planting zone. See what comes up. It's okay if ground cover goes over the drain covers, it can easily be moved or cut away when you need access.

Depends on how much maintenance you want.

Similar side walkway here, but dropped in concrete pavers decades ago and never need to deal with mud, grass, etc. Just a clean walkway front to back.

Naturally, you'll install with a slope away from the home, and towards the front of the home so rain naturally flows towards, then under that fence door then to the curb. Against the side fence, you'll want to slope away from that, too, so the water doesn't run into the neighbors. So looking at the walkway - a slight "v" shape for the slopes.

Naturally, if the neighbor's land is sloped such that it'll dump rain at their side of the fence, you might still get too much water on your side (rain from both sides pooling on your walkway).

If they don't slope and drain properly, only way is to replace that wood fence with a concrete block wall that'll keep their water on their side. (And then instead of a wood fence lasting a decade to a few, a concrete wall that'll last many decades.)

If you have tons of rain, any drain system can help if it tends to pool in one place despite proper sloping. For extremely rainy areas, you might even need underground catch basins. And modify the gutter drain to dump water on the other aside of the fence door, ideally many feet away from the foundation, too.

Also, used to be you needed tons of digging and sand and base, but with plastic paver bases and weed liners, much quicker and easier to get a nice base for the pavers.

I’m trying to go no maintenance so definitely considering the pavements option but pavements need gravel base and I’m concerned about the gravel moving towards the neighbors. Will a metal or stone retaining wall be enough?

Not going to lie - thought this was a /nosleep when I saw the title.

Slip 'n Slide

I’d dig out and remove some dirt in high areas or around basins. Then lay down weed barrier. Then throw down a few yards of black star gravel. Could put some slate stones on it to make a walking path if wanted.

You want to first direct any pooling water away from that area. Even if you do a base and pavers, if a lot of water collects its going to wash away the base and your going to get pavers that will start sinking. As for keeping the pavers retained, a concrete retainer would be best.

Concrete stepping stones and potted plants, if you want any. The cracks between should avoid any excess water going into your neighbor’s yard.

With it sloping away from your house, it's doubtful you need the expense of a french drain.

Throw a bunch of gravel back there and call it a day.

I can imagine some nice rock with stepping stones (no grass). Potted plants/trees lining the house and a few hangers on the fence.

That’s what I was thinking too. It gets really slippery when wet so I was considering just putting some rocks in there.

Imagining the same thing but at a budget.

Yeah, the gravel will allow the water to drain properly. I'd put down some rolls of weed blocker, so nothing grows up through it. Then pour gravel and you're done.

You can get interlocking gravel base, which will keep the gravel from migrating. It also helps you get level surface since the sides allow you to visually identify low spots.

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I threw down some astro turf rolls from Amazon we had lying around. No issues so far. Dog uses it for the bathroom sometimes and it rinses clean and poop picks up easily.

I second that. Gravel is easy and relatively inexpensive.

Would a metal retainer work?

Depends on your climate. In the Midwest we get lots of thaw freeze cycles and that wreaks havoc on pavers if you don’t have a good retaining system. The metal is fine for a bit but then after a few season your pavers start to shift. Concrete works the best if it’s deep enough.

https://www.familyhandyman.com/project/this-paver-patio-base-will-save-your-back/

Look at the photo between plastic paver vs traditional base - the point is you don't need gravel!

But as for the edge, you can do anything from metal to wood to bricks standing on edge embedded flush into the ground to keep the edge straight and from moving about.

Really depends on just how "muddy" and quick that ground becomes...

But if you want to do gravel....

https://unilock.com/storage/2019/11/advanced-tech-guide-en-final-lr.pdf

That's a good reference on what you'll need underneath and to the sides...

I’d bet an elite variety fine fescue would grow just fine. You can get that stuff to grow in complete shade