[Thunk]

A cheapo set of dumbells should do the trick. Stand with suitable weight bells in each hand (you don't really want to become lop-sided, do you?), feet apart as if on the shooting line. Lift with arms straight until arms are horizontal, hold for a count of five, slowly relax until your arms are hanging straight down again. Repeat ten times, and increase the number of repeats gradually - if you do this daily, then increase to 15 repeats after two weeks, and 20 after another two. Also increase gradually the count for which you hold the bells at full extension.
Note: you don't need massively heavy dumbells - it's the repeats that build muscle, not excessive weight in the first place. If you are finding it too hard to reach ten repeats, use a lower weight.

Also - alternate lifts so that the shaft of the dumbells is horizontal, then vertical. It brings different muscles into use.

You could also try the doorframe method; stand beside a doorway, arms length away from it. Put your hand on the doorframe with the arm horizontal.
Slowly collapse the arm until you are leaning with your shoulder aagainst the doorframe (don't move your feet!) Then push with the arm until you are standing upright again. Repeat as much as is comfortable - don't overdo it and strain yourself. Once again, build up the number of repeats gradually.
And do make sure you exercise both arms equally!