Alter arguments in custom command

Well, you can do it with much shorter code using xparse. No need of if then, we simply use the \str_case_e:nnF macro from expl3.

Edit: As Manuel mentioned in the comments, it is better to grab the first item of the input by running the helper without braces. Then C^* also gets converter into \mathbb{C}^*. The obvious reason for splitting into cases, would be if some was to use \mathbb others \mathcal etc.

\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand\funcHelper{m}{
   \str_case_e:nnF { #1 }
   {
     {N}{\mathbb{N}}
     {Z}{\mathbb{Z}}
     {Q}{\mathbb{Q}}
     {R}{\mathbb{R}}
     {C}{\mathbb{C}}
     {K}{\mathbb{K}}
     {F}{\mathbb{F}}
   }{
     #1
   }
}
\NewDocumentCommand\func{mmmm}
 {
    \arraycolsep=1.5pt
    \begin{array}{rl}   
        #1
        :
        \funcHelper#2
        & \longrightarrow
        \funcHelper#3
        \\
        x & \longmapsto #4
      \end{array}
    }
\ExplSyntaxOff

Daleif's answer is very pretty and is a really nice demonstration of the power of LaTeX3! On the other hand, I prefer a minimalist approach:

\documentclass[a4paper]{article}
\usepackage{amsmath,amsfonts}
\usepackage{array}

\newcommand\func[3]{%
  \begin{array}{r@{}l}
    #1\colon{\mathbb #2}&\longrightarrow{\mathbb #3}\\
        x&\longmapsto #1(x)
  \end{array}%
}

\begin{document}
  $\func{f}{R}{C}$

  $\func{f}{R}{C^*}$
\end{document}

This produces:

A few comments:

• The fourth argument in the OP is redundant as it is the same as #1
• The first character in #1 and #2 will be typeset in \mathbb, whether or not this makes sense but this fits with all of the examples in the OP so I think that's OK
• I didn't know about \longmapsto so this question was useful to me!
• It is better to use \colon than :
• I added @{} to the array to gobble some extra space before the arrows

Edit

Using xparse you can have an optional fourth argument, that defaults to #1:

\documentclass[a4paper]{article}
\usepackage{amsmath,amsfonts}
\usepackage{xparse}
\usepackage{array}

\NewDocumentCommand\func{ mmm O{#1}}{%
  \begin{array}{rl}
    #1\colon{\mathbb #2}&\longrightarrow{\mathbb #3}\\
        x&\longmapsto #4(x)
  \end{array}%
}

\begin{document}
  $\func{f}{R}{C^n}$

  $\func{f}{R}{C^*}$

  $\func{f}{R}{C^*}[g]$
\end{document}

This produces:

It is slightly non-standard having the default argument at the end, so you might want to use

\NewDocumentCommand\func{ O{#2}mmm }{%
  \begin{array}{rl}
    #2\colon{\mathbb #3}&\longrightarrow{\mathbb #4}\\
        x&\longmapsto #1(x)
  \end{array}%
}

instead but I would put the optional argument at the end as this looks more natural to me.