Mine Cart Spelunk is one of the events in the Wild West booth area in Free Play. In order to get it you need to unlock the first three booths in order, or you can just play Adventure Mode which will cycle to Mine Cart Spelunk eventually without needing to unlock it.
Mine Cart Spelunk is a shooting game where carts with targets come out of shafts towards you. In the upper area there will be bats with targets as well -- they come in groups. First, two bats worth 200 points each, and then if you shoot both of those, four bats worth 500 points, and then six bats worth 1000 points each.
It's not hard to get 5000 points while the pies are flying, but if you want to set yourself up for the best chance, as soon as the level starts kill the two 200 bats and the four 500 bats. When the six 1000 bats appear, DO NOT SHOOT THEM. They'll stay put up there. Then, when the pies start flying (you'll see a person-shaped cutout in the center of the screen and a pie will start winding up to launch), shoot the bats. Even if you only hit five of the six in time, you've got your 5000 points.