You can do this using the colororder to control the coloring of lines, it has the modulo behavior built right into it. But using a colorbar to represent the colors takes a little extra work, because the colorbar is tied to the Axes colormap (i.e. it's intended for labeling continuous colors rather than discrete colors). It's easy enough to make them line up:
rng(1) % just for reproducibility of the example
% A sample dataset of 36 lines
vals = rand(100,36)+(1:36);
% Note that his is the same as looping over columns and plotting each one
plot(vals,'LineWidth',2)
% A palette of 12 colors
mycolors = rand(12,3);
% Set the colororder to make lines use the palette
colororder(mycolors)
% Fake out a continous colormap for the colorbar
% Using colorlimits is an easy way to make the tick values line up
clim([.5 12.5])
colormap(mycolors)
cbar=colorbar;
cbar.Ticks=1:12;
cbar.TickDirection='out'; % just because I thought it looked a tad better
Alternatively, if you wanted to use a legend to label the lines, it's a little easier:
clf
rng(1)
vals = rand(100,36)+(1:36);
myplots=plot(vals,'LineWidth',2);
mycolors = rand(12,3);
colororder(mycolors)
legend(myplots(1:12),string(1:12),'Location','eastoutside')
