Nice question. I'll admit I am not 100% sure on what follows - there should be something in Adams' book though, I would have thought?
I believe we get that it all commutes with the structure maps automatically.
Recall that a map of spectra $f:E \to F$ is a collection of maps that are compatible with the structure maps. For two maps $f_0,f_1$ a homotopy is then a map $g:E \wedge I_+ \to F$ such that the restrictions to $E \wedge \{ 0 \}$ and $E \wedge\{1 \}$ are $f_0$ and $f_1$. The map $f$ will be a cofibration if it has the HEP for all spectra, and then we can form the cofiber of $f$ as the pushout of the following diagram
$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} E & \ra{} & CE\\ \da{f} & & \da{} \\ F & \ra{} & C_f=F \cup_f CE \\ \end{array} $ where $CE$ is the 'cone' $E \wedge I$
Similarly the fiber can be formed by a similar type of pull back diagram.
It should be true, I believe, that (since everything commutes with our maps. Or in Adams' terminology we can use strict maps on cofinal subspectra)
$(C_f)_n = C_{f_n}$ and that $\Sigma C_{f_n} = C_{\Sigma f_n}$
Ok, the diagram for the fibre should look like
$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} F_f{} & & \ra{} & PF\\ \da{} & & & \da{} \\ E & & \ra{f} & F \\ \end{array} $
where $PF=\text{Map}(I,E)$