Step 1: Cooler
Don't buy one. Get one, with a drain, at a garage sale. Unscrew all the hardware.
Step 2: Drain
Remove the drain with a wrench
Step 3: Drain Cap
Remove this too, and set it aside.
Step 4: Legs
Build four legs, using wood glue and a brad nailer. I made mine 33" tall.
Step 5: Build the Stand
Lay the cooler upside down on a flat surface with the lid unscrewed and removed. Measure the width of the cooler. The easiest way is to lay down the cooler and measure with two legs standing up, and then you have your width. Cut one slat for the width and slide the cooler in to make sure it fits. Since I used 1x4 boards, I had to add 3/4" x 2 to account for the long boards, because they need to fit in there, too.
Then start building the other side. Next, measure the length of the cooler while it is sitting in that position. Cut one slat and see if if fits right. Then add some more. Screw all to the legs from the inside so you don't see any screws. If you screw up, unscrew and redo.
Step 6: Complete Box
Keep adding slats until you have room under the box to put some cross bracing (not shown). Don't put the cross bracing in yet though. You are going to put some framing on top before you fit your cooler permanently.
Step 7: Build the Lid
Put in the four flat slats you see by gluing them and nailing them down. Then flip the cooler and box back over and you now have where your cooler will sit in the box. Now put some cross bracing in to hold the cooler. Next, put the cooler lid on so it fits snugly.
For the lid, simply build the perimeter of the box around the lid. Then put boards on top and nail them down.
Step 8: Finish the Lid
Put some boards across the lid box you just built and glue and nail them in.
Step 9: Add Bottom Shelf
Glue and nail some slats with some space in between to allow for drainage
Step 10: Laser Engrave
Here is a plaque I had laser engraved. You probably don't have capability to do this, so just buy a sign to put on the front.
Step 11: Add a Side Table
if you want
Step 12: Add a Magnetic Bottle Cap Opener
Found it online at Amazon. About $20, but it catches the bottle caps so you don't have to install a catcher.
Step 13: Done
I realize that this is not a perfect step by step, but hopefully you can see that it can be done with minimal tools. I used a chop saw, drill, and brad nailer. Finished it with two coats of spar varnish so I could leave it outside. Had a plumber friend fit a spigot on the outside of the cooler box onto the existing cooler drain. Took a lot of work and fiddling but I think it came out OK.