I found the Neocities site crawler's user agent!!!
Here it is:
Mozilla/5.0 (Xll; Linux x86_64; rv:50.0) Gecko/20100101 SlimerJS/0.10.2
How I did it
I added this page to my website. Then, Neocities posted my update, including a screenshot of the added page. The screenshot contained the screenshot-taker's (a.k.a. the Neocities site crawler) user agent.
What about IP?
I tried it (using GOGO JS which uses www.ipify.org), and then I tried it again (using an iframe that directly loaded api.ipify.org.). Both methods didn't work.