There are at least three ways to approximate the age of your budgie.
Age test 1/3: check the budgie's cap feathers.
Young budgies have bars on their head running all the way down to the cere. When a budgie is about 4 months old, he will go through his first molt. The top feathers on the head will be replaced by feathers that are no longer striped. Therefore you can determine that a budgie with a striped cap is less than 3 to 4 months old, while a budgie with a clear white or yellow cap is older than 3 to 4 months.
Exception! Lutino or Albino budgies don't have the usual feather markings and therefore this age test cannot be applied to them. The same goes for Recessive Pied budgies.
Left: compare the older bird on the left and the younger one on the right
Right: this age test cannot be applied on a recessive pied budgie
Information and pictures (c) budgieplace.com