A creature with a spell-like ability limited by "per day" uses will regain that ability 1 day from the time it used it last. As for spells they work the same as the class of which they are casting from; I.E. A cleric will pray to his god at a specific time each day and a sorcerer needs 8 hours rest and 15 minutes in meditation to get them back.
For your more complicated question about 3/day abilities and it only used 1 of its charges then that charge would refresh after 1 day or 24 hours from when it was used however if it used say 1 charge in the morning and 1 charge in the afternoon and 1 charge in the evening they would all return after 1 day from the last time it was used not the first time.
It is also a loose theory in that all spells don't really set by day but rather 8 hours of rest and 15 minutes to an hour of meditation... By that rule the per day uses would not recover until the creature had 8 hours of uninterrupted rest and then at least 15 minutes to refresh the daily uses of his Spell-Like abilities.
For book reference the best description comes from the Rules Compendium for Spell-Like Abilities stating them to be akin to the Bard or Sorcerer non-prepared spells, which is as follows;
Players Handbook I: page 178
Prepared Spell Retention: Once a wizard prepares a spell, it remains in her mind as a nearly cast spell until she uses the prescribed "effort" to complete and trigger it or until she abandons it. Upon the casting of a spell, the spell's energy is expended and purged from the character, leaving her feeling a little tired. Certain other events, such as the effects of magic items or special attack from monsters, can wipe a prepared spell from a character's mind.
Recent Casting Limit: page 179
As with wizards, any spells cast within the last 8 hours count against the sorcerer's or bard's daily limit.
Spell-Like Abilities: Tome of Magic page 113
Although you do not prepare spells, you must rest for 8 hours and meditate for 15 minutes each day to regain your use of mysteries just as a sorcerer or bard must rest and meditate to regain use of spell slots.
Mysteries are spell-like abilities FYI.
Spell-Like Abilities: Tome of Magic page 139
Spell-Like abilities are magical and work just like spells (although they are not spells and so have no verbal, somatic, material, focus, or XP components). They do not function in an antimagic field and are subject to spell resistance if the spell the ability resembles or duplicates would be subject to spell resistance.