Okay, I found a workaround using xlim, removing the border and making the background white. There's still a little bit of the x-axis behind the colorbar, but it looks decent:
figure(1) %first figure
count2(count2 == 0) = 1; %log(0) not defined - replace all zeros in count
imagesc(count2') %real values, but "wrong" colormap
cbr = colorbar('Yscale', 'log'); %logarithmic colorbar
figure('Color',[1 1 1]) %second figure with white background
imagesc(log10(count2')) %logarithmic image with colormap that fits colorbar 1
set(gca,'YDir','normal') %flip image
a = get(1,'children'); %extract colorbar from figure 1
copyobj(a(1),2); %copy to figure 2
close Figure 1
xlim([1 67]) %shrink imgage, so that colorbar doesn't overlap
box off %rempove frame (that still overlaps colorbar)