I've tried to import my own North Arrow with the Style Manager (ArcGIS 10.1) both as .emf and .bmp but the North Arrow is pixelated although the resolution of the picture should not cause this problem (i've saved the Arrow in a DinA3 file) and although the vetor format of the picture. Has anyone an idea how to solve this problem?
Unfortunately, ArcMap's anti-aliasing capabilities are very limited. (See also the similar GIS.SE question about marker symbols being low-quality after .emf import.) Even if your high-quality symbol still exists, it is displayed crudely on screen.
This problem only arises during low-resolution display (in most cases during display of maps on a screen, but also if you export to low-resolution graphics). It should not occur if your map is printed in high resolution, exported as pdf file, or exported as high-resolution raster image.
The most advanced way to avoid pixelated display of your north arrow / symbols is to use a custom font. (To render TrueType fonts, ArcGIS uses the anti-aliasing settings of your display, which can be configured in Windows.) This is probably tedious if you need to display a single symbol only. There is a question addressing common font editing tools if you want to dig deeper.
Note that fonts are monochrome by design; simple multi-color symbols are possible by stacking different layers within the Style Editor, but symbols with more sophisticated coloring may not be possible.
If you are preparing a map for print or pdf export only, you may probably ignore the pixelated view which misses missing anti-aliasing because it occurs during map display within ArcMap only, but not after print or export to pdf - your full-quality vector symbol is preserved there.
If your final product is a map for low-resolution display only, you can achieve quite good results by embedding your symbol (north arrow) as a raster graphic which has been designed for that particular low resolution. (You could export the custom vector symbol to a low-resolution raster, utilizing the anti-aliasing capabilities of your vector editing software.)
There are lots of cons to all of these methods, particularly (1) the high effort using ttf, (2) the coarse screen display by ignoring the issue, or (3) the coarse display of low-res raster graphics if viewed in higher resolution.
I would recommend the universal method using a ttf (see 1) if ignoring the issue (see 2) is not an option.
Probably you can also optimize the overall display of your .emf vector file. Note that the page size you mentioned is not relevant in order to determine the quality of a vector graphic. Avoid unnecessary vertices, make use of curved paths and choose appropriate line widths depending on the final display size. You may also consider to generalize high-level details in order to improve small-sized display of your symbol.