I have found some videos that can help with web fluid
Real Synthetic Spider Silk
Kevlar Web Shooter
Swinging from one arm can be dangerous, you're going to need a harness of sorts
To crawl up walls, I'd recommend Gecko Tape or Gloves
For safety, you may wish to make wingsuits under the arm incase you need a safe way down, whether that be due to a lack of web fluids or the buildings are too far. The wings can be transparent with black marks to look like webs (You may also need a parachute, again, for safety)
For a "Spidey Sense", I'd recommend a radio-frequency omni-directional scanner. Something like this:
The suit is going to take on a lot of impact, fighting crime and swinging buildings, but also needs to be flexible. I'd recommend kevlar, but not just any kevlar, liquid armor kevlar. It's flexible and non Newtonian
The CPU should be a raspberry pi, to act as a GPS, communicator, processing unit for the spidey sense, and likely more. Spiderman canonically has a belt to help store and carry things, like web fluid. One is likely to need batteries, of coarse, to power the web shooters and processor. I would NOT recommend lithium ion batteries. They're efficient, but high impact could be dangerous
WARNING: I DO NOT CONDONE VIGILANTISM NOR SWINGING DANGEROUSLY FROM BUILDINGS