Monkey bread is a rich bread, not unlike brioche, with a sugar coating that acts as a natural preservative. Although monkey bread contains eggs, this doesn't make the bread more perishable than a cake. The interior of monkey bread stays moist longer than regular bread, thanks to its fat content. It is, however, prone to surface drying if it isn't stored properly.
Never store monkey bread in the refrigerator. During cold storage, between 33 and 40 degrees Fahrenheit, starch reverts to its crystalline stage, prompting hardening and staleness. It doesn't matter if the bread is stored in an airtight package; the hardening is from temperature, not oxygen contact.
Room-temperature storage is your best bet for maintaining freshness if you intend to eat the monkey bread within a couple of days. Store monkey bread in a plastic bag and out of direct sunlight. The bag minimizes moisture loss and keeps the bread soft and supple for about two days.
Freezer storage gives monkey bread the most longevity and freshness retention of any method -- it stops staleness. But it isn't something you want to do if you intend to thaw it and refreeze it; you should only freeze monkey bread one time, as repeated thawing and freezing encourages the formation of ice crystals and therefore sogginess.
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To store monkey bread in the freezer, cut it into pieces you can easily wrap tightly in plastic wrap. For example, if you have half a monkey bread, slice it in half. Wrap each piece tightly in plastic wrap and store it in the freezer for up to one month. Never store monkey bread dough in the freezer -- freezing renders the yeast inactive.
Tips
Reheating bread to 140 F reverses staling. At 140 F, the starch gelates and its crystallization reverts. This works best for "day-old" bread. If the bread is dried, reheating won't save it.