Some timelimited prizes will occationally appear in the shop that you can purchase for real money, though I do not think the picnic basket has been offered - that is the only way to get discovery quest timelimited prizes if you missed the deadline for the quest itself
For future quests/events; if you really want the time limited prize, look up the quest/hobby/event so you know what to do, when, how many sims it takes, what age etc. GwG has an excellent blog. Prepare to spend real money on LP or SP to make sure you get it (if it returns you will have to pay real money anyway).
