how to style multi-value interpolation function plot?

If the sampling is dense enough, as it usually is in NDSolve output, ListLinePlot will do style each curve separately.

ListLinePlot[f]

ListLinePlot[f, PlotStyle -> {Red, Green}]

As already noted in the comments my styleSplitter will handle this, as will xslittlegrass's restylePlot2. If you are after the simplest possible method (least code) this is the best I can think of at the moment:

Plot[f[x], {x, 0, 2 Pi}] /.
  {d_, x__Line} :> {d, Riffle[{x}, {Red, Blue}, {1, -2, 2}]}

This could be hard-coded for only two Line expressions but Riffle is more general.

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It occurs to me that you did not forbid modifying the InterpolatingFunction expression, therefore we could split that in two before plotting:

f1 = MapAt[First, f, {4, All, 1}];
f2 = MapAt[Last,  f, {4, All, 1}];

Plot[{f1[x], f2[x]}, {x, 0, 2 Pi}, PlotStyle -> {Red, Blue}]

In version 9.0, you can also use

Plot[f[x] , {x, 0, 2 Pi}, PlotStyle -> (i = 1; {{Red, Green}[[i++]], #} &)]