If your new tank is up and running, I wouldn't worry about fixing the UGF in the mini tank, just move ol' Hank on up in to his fancy new home
Feed him very little for the first little while, change the water frequently and monitor the levels. Hank will be so much happier he won't know what hit him.
If you still want to fix it, in response to your question of how: I'd just dump the water and gravel into a clean bucket, set the UGF plate on the bottom, and scoop the gravel back into place. You can replace all the water with new dechlorinated water or wait for the water in the bucket to settle and scoop some of it back in and top up with fresh.
Since you were curious, the way it filters is by drawing water through the gravel bed - using the gravel as a filter medium. The bubbles in the plastic tube rise, and pull water up with them. In turn, the tank water slowly passes through the gravel bed, leaving behind all the dirt, and clean water flows under the filter plate, rises up the tube and out the top with the bubbles. Since all the debris is left in the gravel, you have to vacuum it out with your siphon or Hank's digging will disturb it, and the tank will always be full of floating muck. This is one reason most of us wouldn't recommend a UGF with fish that like to dig. Helpful bacteria also build up on the surface of the gravel over time, and use the Ammonia and Nitrite as a food source as the water passes by, so the filter is, theoretically, a biological and mechanical one.