The domain name I want has an expiration date in late July, 2001.

About four months ago. Verisign appears to be the registrar.

Network Solutions does not show it as available when I check, but offers a chance to "back order" it for $49 through "SnapNames" (apparently a partner or part of the same corp.).

I know of the services which check for expiring domain names for a fee. If "SnapNames" is partnered with NS, would they get first crack at the names? Or would another service have an equal chance?

Could I call NS and tell them I want to buy the domain, since it has expired months ago?

Yes I think tthe best way is to contact netsol and you probably want to check http://www.expirefish.com. They will help you grab the domain name automatically when it becomes available, and the service is pretty cheap.

but I'm positive with snapnames you can have the best shot at getting the domain, other services like www.expirefish.com for example only checks and will register if available every 10 minutes, and snapnames is up to the second.