A few things to check.......
Make sure that the Ethernet cable from your laptop connects to one of the three LAN <--> Ethernet ports on the Time Capsule.......not to the Ethernet port labeled "O".
Turn off the WiFi on your Mac to make sure that the connection will be over the wired Ethernet cable.
Does the Time Capsule now appear if you check in AirPort Utility?
If not, does an Other WiFi Devices button appear? If yes, click on that to see if there is an option to select either WiFi or Ethernet for the connection.
If you are still having difficulty with the back up, post back with the exact text of any messages that appear on your Mac. A screenshot would really be helpful as well.
If the backup starts, but does not finish, what message appears?
It's your call, but I would not keep any data on the Time Capsule that you cannot afford to lose, since the Time Capsule power supply and hard drive are long overdue to fail on a 10 year old product.