So all of your devices are on same vlan and your switches are interconnected with 1 upstream connection to this router in the mail room, or do all the switches go directly to the uni router?
If you only have one uplink to their router, you could just ask to have this block they gave you routed to you via a /30 or /31 transit network, then all their switches could maintain their same IP and you would just have pfsense as firewall between their network and the yours. You could then allow them access to the IPs the switches are on and all your devices could maintain their public IPs you would just allow traffic at pfsense.
Or you could nat off your network yes, you could then change all your switches to your private network and port forward to them via a public IP you put on pfsense wan. Since you had a whole block to use.. You could the same switches IP and just forward to the new private IPs of the switches.
The routed network is the cleanest option if you ask me.
Your other option - which I don't like but would be to setup pfsense in transparent mode and bridge. I would not recommend this option, but this is another way for you to leave all the ip addressing the same on your devices and their switches while still giving you a firewall between their network and yours.
As to learning pfsense – just connect pfsense to one of your switch ports. Put another switch behind it connected to pfsense lan interfacde and play with putting some devices behind pfsense. This would be a mini version of option 2.