I do face painting (on cruise ships sometimes) and I can tell you that if you get a small paintbrush, a makeup sponge from the dollar store, and some red and of black Snazaroo face paint you can do it yourself in about 5 minutes. A kit like this and a sponge is all you will need, and you can even put the lid back on and bring it to the park for a touch up later, but I bet he doesn't care too much by then.
https://www.michaels.com/product/snazaroo-rainbow-face-paint-kit-10115421?michaelsStore=1178&inv=1
To do his face, dip the sponge in some water and rub rub rub on the red disk. (Don't get the sticks. You want the disks of face paint) Then wipe the red color over his entire face except for his eye area: forehead, around his nose, his cheeks, and even across his mouth. (You can wipe that off with a wet paper towel.) Maybe do 2 coats - each should dry in moments - but don't forget to moisten the sponge again before rubbing the paint disk. Once that's dry, take the paint brush and some water and swish swish swish the brush across the black paint. Outline around his eyes with black, and just do a web design by making some black slashes, and curves connecting them for a 'web'. You really can't go wrong.
You could also choose to just do the Spiderman paint around his eyes as a mask, especially if he's a kid who doesn't like things on his face. But the Snazaroo on a sponge is so light most kids don't mind it. Here's an example of the latter:
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