In theory, yes, you can do that. However, VLANs can span switches and in a multi-switch, multi-VLAN, multi-routed environment, the IP address will only appear in the ARP table on the switch where the gateway is configured. If the PC is not directly connected to that switch, then the IP address will be associated to the uplink port of the switch with the gateway. This is why I mentioned bouncing the MAC address off of your DHCP leases to find the computer name.
Of course, there is another way to do this. If you know the name of the computer you're looking for, you can always run nbtstat -a <computername> to get the remote computer's MAC address, then search the forwarding database (MAC table) on the switch to find which port it's associated to.
Always work from the bottom up of the OSI model. If you can't figure out it out Layer 1, then move to Layer 2...which would be to find the MAC address. As long as the link is active, the MAC should appear in the forwarding database and show what port it's on. IP's come into play at Layer 3, but see my note above on figuring out which port it's plugged into.
There are other ways to figure this out with higher-end managed switches and their associated management applications, but you'll pay A LOT more for them.