if you want strength do low weight high rep curls if you want bulk do high weight low reps

Like others have said, get a set of dumbbells; you can buy them used from craigslist, they're dirt-cheap in there.
If you just want to work your arms (biceps, forearm, shoulders and triceps is what I'm looking at; if you don't keep a balance between those you'll look funny) and don't have enough time (you said you were busy as hell; this routine should last around 30-45 minutes)

We'll separate this in two different days: Pull (biceps, forearms) and Push (triceps, shoulders). A good plan for these would be Pull on Monday and/or Friday (depending of your urgency and whether you still feel pain from the previous workout. If you do, wait until next week) and Push on Wednesday and Saturday (same rule applies here).

The ideal thing would be if we added Chest to Push and Back to Pull, but you say you don't have much time. Let me know if you want me to add those and I'll write them in another comment.
This routine works best because of cumulative fatigue; it promotes muscular hypertrophy. For this, the rest between sets MUST BE no more than 30-40 seconds.

Look for the names of the exercises in YouTube if you're not sure how they're properly done. (All of these can be done with dumbbells instead of barbells)
When it says "12 reps" you should not be able to complete a 13th rep. If you are, increase the weight. Same applies to 10, 8, etc.

I'll post the routine in another comment, since I would exceed the number of line breaks FJ allows me to use.

Thanks that sounds pretty good actually, i was trepicdent to post here as i thought it would be full of people teling me to do crazy exercise routines, but this is pretty reasonable.

You're welcome, I hope it works well for you.

PULL DAY:
Supinated Curls: 12reps, 10reps, 8reps, 6reps, 15reps with lower weight.
Hammer Curls: 12, 12, 12, 12
Concentrated Curl: 12, 12, 12, 12
Reverse Curls: 12, 12, 12, 12
Wrist Curl: 4 sets to FAILURE (failure is when you're not able to complete one more rep)
Reverse Wrist Curl: 4 sets to FAILURE

PUSH DAY:
Arnold Press: 12, 10, 8, 6, 15
Lateral (side) Raises: 12, 12, 12, 12
Seated Military Shoulder Press: 12, 12, 12, 12
Seated Overhead Triceps Extension: 12, 10, 8, 6, 15
Bench Dips: 4 sets to failure.
Lying Triceps Extension: 12, 12, 12, 12

And that's it! :). Good luck.

you could buy a pair of dumbbells, and just do a weight which isn't too heavy, but its heavy enough to make you feel like you're working your arms after 3sets of 10 reps, and if you don't want to become buff, just don't do it too often.
I'm sure someone will be able to give you more profession advice, but that should be a start for you.

sorry, just saw the guy below post almost the same as me before me :(

If two people are suggesting it he'll know it's the right thing to do :)

Get a set of dumbbells, do around 4 sets of 8/9 reps of a weight that when you get to the 8th/9th rep then you fail it. Take a whey protein shake before and after, and take one shake on the day you don't work out. Do this at least 3 times a week.

That's the bear minimum to help gain muscle, you won't get bigger biceps or any muscle with little effort. Try adding in 4 sets of push ups as well to build your triceps and chest.

** do curls or hammer curls...or alternate between.

**buttholee rolled a random image posted in comment #17998 at Social Board **
i bet the left arm is much weaker

Ummm, that is my left arm...