I know that if the Imperial player puts an Agenda side mission in play and it isn't played, that the Imperial player gets the reward. My question is how long does the agenda mission stay active?

The moment you play a side mission (note: Side mission, NOT story mission) that isn't the Agenda side mission, it gets discarded and the Imperial player gets the reward. The cost of preventing the Imperial player from getting the reward is sacrificing a side mission slot in the campaign that you may rather use for a different mission/reward. In some cases it may be better to let them have the reward so that you don't give up the slot...

The Imperial player may make things harder by playing 2 Agenda side missions at the same time. At that point they will get at least one of the rewards either way, because choosing to complete one will automatically discard the other.

Hoth makes it an even harder choice with Threat missions.

Don't confure Agenda side missions with Agenda forced missions though. There's 2 kinds of Agenda missions. Forced missions occur immediately, the rebels don't have a choice, it can occur at any point in the campaign, and it does NOT replace any of the mission slots. It gets written into the seperate area for Forced missions.

The moment you play a side mission (note: Side mission, NOT story mission) that isn't the Agenda side mission, it gets discarded and the Imperial player gets the reward.

The Imperial doesn't gain the reward until you finish whatever mission you decided on.