My thought process is to get fermentation started, then once the first two days of fermentation has completed. Filling a cooler with ice and water. Then submerge the little Brown Keg (Mr Beer Keg). To keep it at the Lagering temperatures. Any more advice is very appreciated.
I believe the Mr. Beer is a 1 gallon size and fits in most fridges nice. However most fridges work between 35-45°. Your target range for lagering is 48-54°F.
With out electronic temp control, you will literally need to babysit the fermentation. But you may be able to pull it off with a simple temp strip on the fermentor. By placing it in and out of the fridge every few hours, but don't let it warm up past 58°. Getting too cold really won't harm the beer, it will just slow down the fermentation.
It's a lot of work for so little beer. I would suggest sticking to ales until you expand your homebrewing more. Usually lagering is one of the last styles homebrewers perfect, because of the need for critical temp control.