4e clearly wasn't focused on enabling sandbox play, so you'll need to do more thinking, but it should work fine eventually. Something I find mildly aggrevating is the lack of guidance on skill DC's. Most skill DC's are easy/medium/hard - which is an inherently relative scaling, so you'll need to make up some more absolute scale here. This is a bit of work, but it's doable. You can make use of another game to help here, too.
Also, I think MM3 monsters still don't deal
enough damage
. And they have too many hitpoints. In sandbox play that's particularly annoying because gameplay is focused more on scouting and choices and less on combat; that means combats need to be resolved quicker - hence you want more damage and fewer hitpoints for monsters (this may make minions too powerful).
I don't think the 5-minute adventuring day is really a problem. There's still an incentive to save dailies since
you never know; and in any case, 3e has this issue much more. And isn't it merely believable and reasonable that a party is more competent and strong if they have just one combat a day than if they have 10? In a sandbox, this works perfectly - they can move quickly through less dangerous regions, or choose to rest extensively after each combat in more dangerous regions. More flexibility - good!
Another issue you'll want to be aware of is that escaping and scouting are harder in 4e. Stuff like dimension door simply don't exist, and Wind Walk and kin are also hard to get. In 3e, monsters moved quickly
in combat, but out of combat (and while escaping), most PC's had the upper hand. In 4e, it's hard for a PC to escape. A few good ideas and some preparation can work wonders here - for that matter, there's no reason you can't reintroduce some of the old staples. An Elixir of invisibility lasts for long enough to escape, and blindsight and kin are fairly rare. An exodus knife could provide a safe refuge until the horde of goblins passes. You could choose to include spells from other games as in-combat rituals to help escaping.
You should also have an idea of how you expect them to scout. Stealth is quite risky, particularly without and adequate means of escape. Divination is often expensive, useless, or unknown. That's probably intentional, permitting the DM to not have to be aware of all the various tricks PC's can use to bypass a BBEG's defenses (so no more 115 abjurations needed to avoid the scry-teleport-instakill of 3.5), but you can of course feel free to add such abilities back into the game. Familiars work particularly well as scouts. Some (such as the cat) don't have a range limit, and you could house rule the range limit to be higher than usual.
If you have some reasonable consistent skill DC's, enable scouting and escaping, and finally keep combats a little quicker than 4e default, then 4e should work excellently as a basis for a sandbox game - better than 3e or 2e in the sense that on-the-fly challenges are easier to run.
