New sidewalk and fence, which first?

That makes the most sense. Every try to pour and finish concrete next to a wall or fence? PITA.

You're doing the engineering for the neighbor's new fence? Or are you just being nice and putting in a privacy fence because they don't like your chain link?

In any case, wood fence posts should never be buried in a concrete sidewalk. The posts will rot out in 10-20 years, and be a PITA to switch out. Your fence contractor should have, or be able to get from his supplier, some sort of socket or stanchion device to put in to the concrete, and the posts then go in or over that. Note that if it is sockets, there has to be a drain at the bottom, and gravel in the bottom of the hole. Other alternative is to set lengths of thickwall galvanized pipe in the sidewalk, and end-drill the posts, and drop a standoff and the posts over them. Simpson is the brand name that comes to mind, but it has been a few years.

Any way you can get the fence guy and the concrete guy together for coffee at the kitchen table, with an accurate diagram of how it is all laid out? Whoever does the cost quote is going to want to come on site first anyway, and I have found that tradespeople (versus sales assholes) are usually quite happy to work this stuff out in advance, since it makes their job easier. With narrow sideyards like this and sidewalks, drainage is always an issue. You don't wanna flood neigbor's yard (especially if they are whiners), but you also don't want to pond water by your foundation. If this is a narrow urban lot and the new sidewalk is within a couple feet of your foundation, you will need to take some precautions with slope, and maybe even a gutter and collection box for the runoff.

aem sends...

. I don't see how they are going to pour the posts for the new fence

Have a look at these Simpson post bases..you put them right into the newly poured concrete and build the fence later