Adding 20 pounds in a month is possible, 5lbs a week on a bench press is actually rather common, but it heavily depends on your background. If you've been working at your bench press intensely for a while and you are where you are, then barring some very poor programming on your part, it's going to be extremely hard.
However, based on where you are, it sounds like you e.g. recently tried to see what you could bench, realized it was far off from the test, and are now in scrambling mode?
Next, men gain upper body muscle strength much easier. If you're female, then this will be much harder.
An example program I might do in this situation is bench press 3 days per week.
- Day 1: Focus on getting stronger. For instance, lift in the 6-10 rep range, minimum 40 work reps (that would be 4-7 work sets)
- Day 2: Focus on endurance. For instance, do some sets of 20 reps.
- Day 3: Practice the test
Minimum one day off between days.
The idea here is you have one day where you get used to lifting a weight heavier than the test weight. This often gets overlooked, but it's simple: if a person can bench press 100 pounds for 1 rep, but improves that to 120 lbs, then 95 lbs is going to be easier.
You have another day where you get used to lifting more reps than the test day. (You're trying to push your endurance.)
Finally, you have a day where you practice the actual test.
Remember though, these are days you are going to have to push yourself. The point is to get stronger / lift more weight, do more total reps, and do more reps at the test weight.
Lastly, if you feel that good about the run, you could consider gaining weight, which will make the bench press easier. The tradeoff is everything else will get harder, but that might be a worthwhile trade at this point.