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The parching touch spell from Sandstorm has a duration of instantaneous but says, "You can use this melee touch attack up to once per caster level."

So how much time do I have to make these attacks? Can I cast it at the morning, and use the touches when needed?

Taken all it once, the spell may seem contradictory. However, what it actually means is that the caster can cast the spell parching touch and thereafter make a number of parching touch attacks equal to his caster level, but the effects of those parching touch attacks remain forevermore—until healed or removed—, even after the caster makes his last parching touch attack.

See, the 1st-level Sor/Wiz spell parching touch [necro] (Sandstorm 118-19) has the entry Duration: Instantaneous, but the spell also has the entry Target: Living creature or creatures touched (up to one/level). Duration on Subjects, Effects, and Areas says

If the spell affects creatures directly (for example, charm person [or, for our purposes, parching touch]), the result travels with the subjects for the spell’s duration [which, in this case, is instantaneous—essentially forevermore until its effects are somehow healed or removed]. If the spell creates an effect, the effect lasts for the duration (176).

This is why the spell's description must say, "You can use this melee touch attack up to once per caster level." (Not incidentally, this way of carefully crafting touch spells that deal damage is also what makes the spell shivering touch subject to DM oversight.)

While a caster could, in the abstract, cast the spell at the day's start and adventure in such a fashion, touch spells make impractical long-term buffs. First, buried in Holding the Charge is the horrible accidental discharge rule that make this unpalatable: "If you touch anything or anyone while holding a charge, even unintentionally, the spell discharges" (Player's Handbook 141 and, yes, that's the whole thing… whatever it means and this DM suggests ignoring it). Second, and more importantly, the same section says, "If you cast another spell, the touch spell dissipates" (ibid.).

Thus it's possible if a wizard is a parching touch specialist for the spell to be his all-day jam—casting no further spells but still parching the crap out of foes until he exhausts the spell's uses—, but I suspect such wizards will be rare.

Technically indefinitely, though various factors make that impractical.

If a touch spell is not successfully discharged the round it is cast (generally because you miss your touch attack) then you are considered to be Holding the Charge and can continue to make touch attacks each round until you successfully hit. You can hold the charge of a touch spell indefinitely, but touching anything or anyone, even unintentionally, will discharge it, making the prospect impractical for any real length of time. You also cannot cast any other spells without losing the touch spell, which is likely to outweigh any benefit you would get from casting it early and holding the charge all day.

Depends on the Spell

Most spells with an instantaneous duration will include an explicit or implicit statement of how long the effects last.

For example the spell listed in the question has the line:

You can use this melee touch attack up to once per caster level.

In other words, the spell instantly has an effect, but the effects last for one melee touch attack/caster level.

Compare this to the spell description for Fabricate taken from the SRD, which has an instantaneous duration:

You convert material of one sort into a product that is of the same material. Creatures or magic items cannot be created or transmuted by the fabricate spell. The quality of items made by this spell is commensurate with the quality of material used as the basis for the new fabrication. If you work with a mineral, the target is reduced to 1 cubic foot per level instead of 10 cubic feet.

You must make an appropriate Craft check to fabricate articles requiring a high degree of craftsmanship.

Casting requires 1 round per 10 cubic feet (or 1 cubic foot) of material to be affected by the spell.

Interestingly enough, the last line overrides the "instantaneous" duration, but more importantly there is no mention of how long the effects of the spell last. This creates an implicit duration: forever.