Yes, you can run a LAN speed cable from your firewall/router/whatever down the hall to the other office's firewall/router/whatever. If they are close enough, consider wireless instead. In office buildings, cabling is run up in the ceiling tile. If this is a long term solution, consider running a real wall jack on both ends of the cable run.
Let us assume the following (because we do not know your equipment): Your office network is a 192.168.1.0 /24 network (192.168.1.0 -
192.168.0.255) Their office network is a 172.16.0.0 /24 network (172.16.1.0 - 172.16.1.255) Both offices have a managable router (Cisco) with a spare ethernet port. Your desktop computer IP address is 192.168.1.25. Their desktop computer IP address is 172.16.1.25.
Connect the two ethernet ports with a crossover cable. If you do not have a crossover ethernet cable or do not know what this is, put a cheap little pocket sized ethernet hub/switch between the two company routers.
On your network router (assuming the port to the other company is FastEthernet 0/1), configure the following: interface FastEthernet 0/1 description *** Ethernet to other office *** ip address 10.11.12.13 255.255.255.252 ip access-group 123 in no shutdown ! ip route 172.16.1.0 255.255.255.0 FastEthernet0/1 10.11.12.14 ! access-list 123 permit ip host 192.168.1.25 host 172.16.1.25
On their network router (assuming the port to the other company is FastEthernet 0/1), configure the following: interface FastEthernet 0/1 description *** Ethernet to other office *** ip address 10.11.12.14 255.255.255.252 ip access-group 123 in no shutdown ! ip route 192.168.1.0 255.255.255.0 FastEthernet0/1 10.11.12.13 ! access-list 123 permit ip host 172.16.1.25 host 192.168.1.25
There is a better, more complete way to do this with policy based routing and NAT but I am leaving this answer for now to keep this simple.