Weird Questions - Would putting a server in the same datacenter as another webserver provide sub millisecond latency between both devices? And is there a cheap method to determine physical data center location of a webserver?

I'm designing a service that will require a low latency (sub millisecond if possible) connection to a few different webservers. Very simple service; will essentially be sending GET and POST requests at a low rate (<1Mbps) to the servers and analyzing their responses.

The only critical aspect to the service is that latency between my server hosting the service and the webservers in question is as low as possible. Like, averaging under 1ms if possible.

I know from a IP lookup that the sites in question have servers hosted by Cloudfare in Ashburn and San Jose. However both of these areas are an absolute clusterfuck of data centers.

Obviously just getting my server setup within a few miles would already be good. But I'd like to get my device physically in the same building.

Is this possible to find out with any existing services? Would it actually provide me the latency I'm hoping for, or is this an impossible task to begin with?

Thanks!