If the Ping from the Mac times out then that explains why the Mac does not see the printer. But note that you never see a printer when adding it via the Add Printer > IP pane. A printer will only be seen if it supports a broadcasting protocol like Bonjour and if it does it will appear in the Add Printer > Default pane.
So working on why the Mac cannot ping the printer, the most likely is the Mac is set to a different subnet to the printer. If you make a note of the IP address used by the Mac, the PC and the printer, they should all be set to the same IP subnet. For a home network this is typically the first three sets of numbers in the IP address. The last number is known as the network address, and each device must have a different network address.
So if the PC and printer have an IP address of 192.168.1.x, where x is the individual network address for each device, then the Mac must also be set to 192.168.1.x.
The other thing to check is the Gateway / Router address. This will be the IP address of the Linksys router and this should also be set in the Network settings on the Mac.
If the Mac is correctly set then are you able to ping the Linksys router or the PC?