For a fan fic is it possible that a fair skinned child with red hair
and negroid features born to a white red haired mother and half negro
father be able to darken his skin by exposure to the sun for months or
years?

Skin color is inherited separately from other traits.  It is not
related to other characteristics like eye color or height or hair
color.  A "dark" skinned  and a "light" skinned person may have
offspring with a range of skin tints.  The offspring are generally no
darker than the darker parent, but can be much fairer.

Anyone can deepen the tint of his or her skin.  We do it all the time
when we get a sun tan.  Unless a person has a very fair skin, thereby
burning in the sun,  skin color can be darkened merely by getting a
tan.  If you stay in the sun year round, you will have a more or less
permanent tan.

Skin color is called a polygenic trait, meaning that more than one
pair of genes are responsible for it.  More on polygenic traits may be
found at:

http://highered.mcgraw-hill.com/sites/007246268x/student_view0/chapter7/

Alanna-ga