Fun fact: Thirty minutes is plenty of time to prep for a game if you're focused on what you are doing. Another fun fact, at one game a week all that means is you need five minutes a day to prep for game and you should have something to go for the next session. How does it work? You leverage plot time vs. player time!
In every gaming session there should be a certain amount of movement on a plot or plots. I call this plot time. For example, when the PCs get a letter from one of the big bads, that is plot time. On the other hand, when the PCs go off to execute two of the three parts of their plan that is not plot time, as it is the players filling the gap. Most game sessions have a lot more player time than plot time, and that is a good thing. It means the players are driving the action, and that is something we want. This is also why what is in your notes as a one line "the players drop off the crystal at the local tavern" can turn into 3 or 4 sessions of shopping and side plots.
Plot Time Needs Broad Strokes Planning
The fun thing about RPGs is that the more you get into the details of a plot item the more you are taking away from your game. Now, it's good to have an idea of how things are happening but you don't necessarily want to be married to the idea.
For example, you need to know that the Black Knight has taken the castle and will trae it back to the PCs if they beat him in a duel. You need to know that the Black Knight is treating the serving staff well and keeping the place organized. You don't need to know the specific details of how the Black Knight is doing that, and even if you did it is probably already in your head if you know your NPC properly.
Little Time and Broad Strokes
So now we have very little time (5 minutes) and we know we only need broad strokes. To do this, we start with the biggest of the broad strokes.
Take your first five minutes, consider where your game is, and where you want the plot to go in the biggest strokes you have. You don't need to go crazy here, but something like "The PCs are hired to retake Castle Ysgard from the Black Knight and his army of Highway Men."
With your next five minutes you're going to decide what three things that story line entails. It should be fairly obvious in most cases. 1) PCs need to be hired. 2) PCs need to meet the Black Knight & Highway Men somehow. 3) PCs need to retake, or fail to retake, the Castle.
With your third, fourth, and five five minute period you're going to ask yourself the most relevant question for each of those three things.
1) Who hires the PCs? The son of the man who lost the castle.
2) How do the PCs meet the Black Knight/Highway Men? The highway men are intercepting dangerous looking travelers on the road.
3) How can the PCs re-take the castle? Anyone who wants can duel the Black Knight for the castle.
The Last Five Minutes
With the last five minutes you simply need to assign a danger level to any part you think there will definitely be conflict. For our three things that means the Highway Men (moderate danger) and the Black Knight (high danger.)
Anything with low danger should be able to hit your average player about 1/3 of the time. Moderate should be about 1/2 the time. High should be about 2/3 of the time.
The Rest is Improv
In truth those three points you made could take multiple sessions, or just one session depending on your group. You will need to be quick on your feet with some of the matters, but that is a skill all GMs should train anyhow. Still,with this you have a plan for three major plot items and room for the PCs to throw in their own kinks.
How Your Session Will Look
You begin your session at the end of last session. Link them from where they were to where they are, but let them tell you why (their characters after all.) From there you can segway into your first point, someone hires the PCs to do a job. This will involve some RP, some checks, and PCs generally will add things. They'll possibly want to follow the person home to make sure he is on the up and up. They may want to rob someone. They may start a brawl. All of that is fine. The point is though, they have a job, a pending payment, and a place to go.
Once on the road you can fill in some of the time with whatever whimsy hits you. Work your way to the second plot item (in our case, meeting the highway men.) Depending on your mood, is it violent? It is peaceful? The highway men are intercepting dangerous looking travelers. This means they could be doing a number of things but most prominently they could be 1) chasing dangerous people off, or 2) hiring them to increase their band.
Either way you can use this to lead to a combat. How? Well, if they're chasing people off you have a fight right there. If they're being hired, the PCs can be given work to waylay a merchant on the road which will also start a fight. Either way you have something to do to let people show off their stats. This can be a very good thing, and if nothing else combat both eats time and is something PCs often want to happen at least semi-regularly in game.
From there you have the end game. The PCs need to find/get to the castle, they meet the Black Knight, and they resolve the issue. Now we've planned one way for the PCs to accomplish this task, but there should be others. If the PCs come up with a good idea, go with it. It will probably work out better in the long run.
Once done, well, you've just planned a session (or two) out with five minutes a day by letting your players fill in the gaps between the broad strokes. I don't recommend only doing a game this way, but when you have little time it is good to at least have something ready to go!
Two things I assume you know when using this method. The first is you know the system well enough you can wing a small to moderate scale encounter. The second is you know your PCs well enough that you can scale said encounter difficulty wise to them with little effort. If neither of these are true, you're going to need more time to fine tune the enemies.