If you are looking to learn ANY concerto in a week and pull it off like a monkey performing a stunt then I would suggest that you retire from the piano immediately. Although, this impression that I am getting may be a result of the replies to your question, so in all fairness here is my suggestion.
If you need to learn the notes to a piece fairly quickly, your main problem is going to be memory. I would advise first looking at the piece away from the piano and dividing it into it's formal functions (ie. main theme, transition, subordinate theme, cadenza, individual theme types of the development (whatever they may be...ie. period, sentence, compound sentence, rarely small ternary) ). If you create "goal posts" by dividing into LOGICAL sections your memory will remain solid for much longer than the day of the performance.
As for the concertos. I have played all 5, but I by no means consider myself to be an expert. However I have witnessed many people quickstudy the concertos, so here are my observations:
Concerto # 1 : second composed of the five. It is more mature than the second, and slightly more difficult interpretively. Beethoven extends the general sonata form much more, and therefore the passages are more complicated.
Concerto# 2: certainly the simplest in structure (easy to memorize). The passage work is slightly easier than in the first concerto simply because there are less passages to master.
Concerto# 3: The first movement is a breeze technically, but musically it is also slightly more expanded than the first movement of #2. If you choose this work, just beware of the second movement. I would read through it prior to starting to the concerto from the first movement (if you work from beginning to end that is). It is very difficult to do this movement well.
Concerto#4: Just don't even try to do it in the time you have. It would be an insult to Beethoven.
Concerto#5: The only problem with learning this one so quickly is its length. It is quite a massive concerto, and require a fair bit of stamina, especially once you reach the third movement. I would avoid this one due to its technical difficulty and depth. It is the most complexly structured of the 5 concertos, and there are consequences performance-wise that result from this fact. It would best be studied under the guidance of a good teacher over a longer period of time. If you are already beyond having a teacher, it is still best studied over a longer period of time. One must digest this concerto, otherwise it would be VERY risky to play...keep in mind that it is the best known amongst the public, and if it is for a competition, the adjudicators will write you off instantly if you don't play this 100% convincingly.
Good luck, and I wish you all the best!