To start with, If you are playing through the 3rd game Year of the Dragon you will not miss this achievement as it's story related.
To launch the transporter balloon you need to get to the end of each of the 5 level in the Sunrise Spring World and go through the return home portal in each. (You don't need to 100% complete the levels you can rush through them and get to the return home portal and that's enough.) Once you've done this, the main characters in each of the levels will be in the balloon room (at the end of the world in a big circle room) and will pull down the floating balloon which you can then ride to Midday Gardens.
If you can't remember which levels you've done you can use the guidebook to figure it out go to the balloon room and see who's there. The main characters in each level are:
Sunny Villa - Mayor Leo (looks like a weird little lion that stands on 2 feet)
Cloud Spires - Cumulus (flying pig thing with spikes on it's head)
Sheila's Alp - Sheila (kangaroo) (you have to buy the portal from moneybags for 300 gems)
Molten Crater - Rocky (rock guy with a mohawk) (the portal is unlocked when you collect 10 eggs)
Seashell Shore - Snappy the Seal (seal that stands on 2 feet) (the portal is unlocked when you collect 14 eggs)