How to find a domain based on the IP address?

Use nslookup

For example, let's find the domain for 207.46.19.254

C:\>nslookup -type=PTR 254.19.46.207.in-addr.arpa   
Non-authoritative answer:                                                   
254.19.46.207.in-addr.arpa      name = wwwbaytest2.microsoft.com            

Note that you reverse the order of the four numbers and append .in-addr.arpa

Remember that an IP-address may have multiple domains, and that the administrators do not always (but mostly should) set up the reverse mappings in DNS.

Two things you can do. One is reverse DNS lookup.

dig -x x.x.x.x

You can also use geoiplookup to find the general area of the source.

Wouldn't the ping -a command also work?

That is, ping -a insert IP address here. It's not always successful though, but it's likely the easiest method.