Should I settle on this salt?

yes settle. That is a great spot. I would've settled it awhile ago if I had the happiness. It's not just 4 fish, but 1 more salt, cow, and horses.

I agree with settling the salt as well, with that many fish you will rapidly get boats, lighthouse, harbor, seaport. A fully-improved fish far exceeds the salt in potential yield. This might be offset by losing inland tiles which will eventually hurt size down the road, but in this case, both look about the same as far as total potential tiles. Actually moving might give you less.

I think that it's early enough to settle a new city there. Personally, I probably would have settled there a lot earlier, but whatever. If 3 of the 4 following conditions are met, then I'd say go for it:

1.) You need a naval base. It looks like you don't have much (if any) naval presence, but it also looks like a pangea map (so you may not need a navy at all). 4 fish will turn that city into a naval powerhouse once a Lighthouse and Seaport get built, especially if you can get the God of the Sea pantheon there as well.
2.) You need the range of a coastal trade city. Resource diversity from that city isn't going to grant you much income from trade routes, but if it opens up sea routes to profitable cities across the ocean, then go for it. Be sure to build/buy a Harbor!
3.) you are currently selling away all your excess salt, and there are still other civs that are willing to buy salt from you OR you need the extra horses.
4.) You can spare 2 trade routes to send both food and production to that city in order to make it productive.