First of all, that switch is a Layer 2 switch, not a Layer 3 (router) switch, so it's not going to be able to do any routing at all let alone static routes (which is what I think you were implying in your question, and wouldn't work anyways).
Secondly, what you're asking for is commonly called split DNS: example.com resolves to an internal IP address when the request originates on the "inside", usually against your internal DNS server.
If you don't have an internal DNS server, your edge router may be able to do DNS proxying and in turn may allow you to setup a static DNS entry (essentially acting as your DNS server). However, alot of consumer-grade router/firewalls will assign the DNS servers the router gets from the ISP, which doesn't really help in your case, as your machines on your LAN would be querying the ISP's (or whatever you have entered in as your DNS servers in your router) DNS servers directly.
You could get away with editing everyone's hosts file and adding the IP and the domain you want to resolve to it, but that obviously gets a bit harder to manage the more machines you have.
I'd highly recommend setting up a DHCP/DNS server internally; it'll give you more control, allow you to do caching (making the Internet feel quicker for regularly-querying DNS records), and split DNS as I've described. Any old workstation will do running Linux of your choice and DNSMasq.