I'm currently using 800x480 resolution and FitViewPort and I divided that by PixelPerMeter variable which is equal to 10 and that helped to move my box2d body faster. But when I tried to use fonts I could easily see pixels. What am I doing wrong and what can I do to fix that ?
Fonts aren't drawn using Box2D so the PPM is irrelevant here.
If you need device-dependent resolution to ensure maximum quality, crisp fonts, you could use a ScreenViewport positioned over your FitViewport, and render all your font related output onto the ScreenViewport.