A creature's caster level and a creature's caster level for specific spells it casts can be different
A creature's caster level is typically set by the creature's class level in a spellcasting class, but effects sometimes change a caster's caster level temporarily upon the caster casting a spell. These temporary changes in caster level don't have any effect when the creature's not casting. For example, the general feat Spell Thematics, in part, says that "you may designate one spell you know per spell level as a thematic spell and cast it at +1 caster level" (Player's Guide to Faerûn 44 and emphasis mine), but you don't otherwise have +1 caster level when you're not, at the moment, casting thematic spells.
Further, the Player's Handbook on Prerequisites says
Some feats have prerequisites. Your character must have the indicated ability score, class feature, feat, skill, base attack bonus, or other quality [like caster level] designated [by the feat's prerequisite entry] in order to select or use that feat. A character can gain a feat at the same level at which he or she gains the prerequisite.… A character can’t use a feat if he or she has lost a prerequisite. (87 and emphasis mine)
And the Dungeon Master's Guide on Prestige Classes, in part, says, "If a character does not meet the requirements for a prestige class before that first step, that character cannot take the first level of that prestige class" (176 and emphasis mine).
With all this in mind, the window to take advantage of temporary caster level increases is open for feats but only just, and it isn't open at all for meeting a prestige class's requirements.
For example, Sllepsstsac is an otherwise normal human wizard 1 who has in her spellbook the 1st-level Sanctified spell divine inspiration [div] (Book of Exalted Deeds 96) that has only sacrifice as its components. She also possesses the feat Primitive Caster that, in part, says
This feat only functions when you cast a spell that does not already have a verbal, somatic, or material component. For each component you voluntarily add, you increase the effective caster level of the spell by +1. You can only add a component that’s not already present in the spell. (Frostburn 49 and emphasis mine.)
Sllepsstsac is caster level 1. When she casts the divine inspiration spell and adds verbal, somatic, and material components to that spell, she increases by +3 her caster level (to +4 total) but only when she's casting that divine inspiration spell. When she's not casting, Sllepsstsac's caster level increase from the Primitive Caster feat is moot because the feat "only functions when you cast a spell." In most campaigns, I suspect that Sllepsstsac typically can't pick the item creation feat Craft Wondrous Item (92–3) as a wizard 1 only on the basis of her having the divine inspiration spell in her spellbook and possessing the Primitive Caster feat (but see below).
Meeting the prerequisites of feats with a temporarily increased caster level
A player could spin a tale that at level 1 during character creation the PC simultaneously cast at a higher-than-normal caster level a spell and took a feat that has as a prerequisite a higher-than-the caster's-normal caster level. (As per Character Creation (7), class features like the wizard class feature spells (56–7) are gained before feats, after all.) Yet, even if the player's tale is a Shakespearean masterpiece, the PC could only use that feat while the PC is meeting its prerequisites: "A character can’t use a feat if he or she has lost a prerequisite (87). The feat's still there on the character's sheet, so it can functions as a prerequisite for another feat or a prestige class requirement, for instance. (To be clear, most item creation feats in particular acquired "early" this way typically can't be used for their own purposes: A creature's often too busy casting a spell to also create a magic cloak.)
After level 1, a creature that simultaneously casts a spell at a high-than-normal caster level and gains experience points sufficient to advance a level can use that high-than-normal caster level to meet the prerequisites of feats, but it's complicated. While the Dungeon Master's Guide on Experience Awards, in part, says, "When the party defeats monsters, you award the characters experience points (XP)" (36), it also says that it's normal to distribute XP at a session's beginning or, more vocally, at a session's end (q.v. 18, 19). Thus, to meet a feat's prerequisite with a higher-than-normal caster level spell in this way the DM typically must end (or begin) the session and distribute XP when that creature's casting that spell at a higher-than-normal caster level. Needless to say, this can be difficult to game and probably possible only under laboratory conditions or if arrangements are made with the DM. Also, like above, because the feat's prerequisite is only met temporarily, the newly acquired feat may remain largely inert on the creature's character sheet for a while.
Meeting the requirements of a prestige class with with a temporarily increased caster level
Using a temporary caster level increase to meet a prestige class requirement is, I think, typically impossible because prestige class requirements must be met before taking a level in a prestige class. A creature that meets a prestige class's requirements only when casting a spell hasn't met those requirements before casting that spell and also doesn't meet them after.
