A quick search of “float switch” found loads of examples, and this site looks quite helpful.
There are numerous styles of switch, and it seems to me you could mount some like the pic inside a vertical pipe open at bottom and top. The switches could be say 100 or 150mm apart vertically up to say 1m, and as the water rises in the street (and thus in the open pipe) the switches trigger in turn. The wires would all run up the side of the pipe to the Arduino on the top (in some kind of waterproof box like a Tupperware), and each switch goes to a different input.
You would know the depth of the water (within the resolution of the distance between the switches of course) at any time, and can also, with the use of some timing and comparisons, know how fast it’s rising or subsiding.
If I were you, I’d get hold of a 44 gallon / 200 or whatever litre drum, a length of drain pipe, and 3 or 4 float switches. Drill the pipe to take the switches, and stick the pipe in the drum. Hook the switches to some Arduino pins, and fill the drum as a test-bed to see how this whole thing works.
That will at least show your proof of concept.