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.