I ran an open-world game once. It kicked my butt, and I don't think I'd ever want to do it again. That's because I'm the sort of GM who always preps as little as possible, and an open-world campaign naturally forces more prep than a linear campaign. That said, my innate laziness did lead me to come up with some tricks for minimizing my prep time.
All Roads Lead to Rome
Unless you have very specific ideas about what each dungeon in each part of your world is about, you can simply prep one or two dungeons and one or two towns, and use them as whichever dungeon or town your players happen to be headed towards. As long as you always have one town and one dungeon prepped that your PCs haven't seen yet, you can use it to fill whatever gaps you want. This keeps you from feeling like you wasted prep on a part of a world your party never explored, and it makes your party feel as if whichever direction they go, you'll have something ready for them.
Even if you decide that this is a cheap trick, and you want your world to be more set in stone than that, having a dungeon or two in your back pocket is a really good idea for when your party inevitably blunders out of the area you've planned.
Narrow Their Options
Sometimes you can't get away with throwing any ol' pre-prepped dungeon in front of the party. Maybe they're deciding between the Mountains of Doom or the Spider mines, and you want their choice to have obvious consequences. In cases like this, you should always try and have the important choice occur near the end of a session, so that you have a whole week (or month, or whatever your interval is) to prep the path they chose.
If possible, you also want to carefully manage the number of choices available to the party. If worst comes to worst, it's easy enough to prep both the Mountains of Doom and the Spider Mines. But if the party's choosing between the Mountains and the Mines and the Forest and the Bay and the Moon Palace, well, you're not going to be able to prep all that, and even if you do, 80% of your planning is going to be wasted.
How you choose to limit options is up to you. You can do it using terrain and travel times (the Sea blocks your way to the West, it's too far to walk to the south), or by managing what information the party has (they've heard wondrous rumors about the town to the north, an NPC has offered to give them a ride to the town to the east). However you do it, giving your players a clear set of choices will keep your prep time lower and give your players a sense of purpose and direction which will energize your game.
Stay One Step - and Only One Step - Ahead of Them.
As you run your sandbox, you'll begin to get an idea for what motivates the characters in your party. This will allow you to more accurately predict which of your plot hooks they'll bite on, and let you feed them information that is likely to get them moving towards something you've planned. Once you've gotten them to a bit of the world you've planned, you've just bought yourself a whole bunch of time.
While your party is busy crawling through your latest dungeon, or running around your latest town, you have a few sessions to plan for where they might go next. Use your knowledge of the players, and treat their current location as a hub from which they might go to two or three different places. Then plan those places. Repeat as needed, and soon enough your players will have built a large part of your campaign world for you.
What it All Comes Down To
Basically, limit your prep by only planning for things the PCs are likely to see in the next couple sessions. Leave the rest of the world vague. If you run the campaign well, you'll always have a bit of warning before your PCs tumble off the edge of the map.
