Select "Fixed World" in the main menu. The objective is identitical Random World, however this time levels and present/power-up locations will always be the same in every playthrough.
Use this guide to help navigate levels:
Another very important tip that will make this playthrough considerably easier is to grab (but NOT use) the present second from the mailbox in level 2. This is a randomizer present which will mess around with all the presents in the game and undo the progress you've made in identifying presents. I recommend taking this present to the wise "carrot" man (first appears in level 4) and having him identify it for 2 bucks, then discarding it.