Welcome to the Forum !
The SLC-5/05 Ethernet to Serial passthrough feature requires you to enter the IP address of the computer that will be using the passthrough channel.
When you enable Passthrough, you specify a Data Table number to use as the passthrough routing table. It can hold one or more IP addresses; if it's not large enough to add your second PC, you'll have to expand it offline and download to the controller.
The Routing Table is configured in RSLogix 500, under Channel Configuration -> Channel Status -> Routing Table.
Or be sure your new PC uses the same IP address as the old one and passthrough should work as-is.