when i took vi-q-tuss as prescribed by doctor, I started to grow rashes. The rashes has disappeared, but now i feel nausea, stomach upset, dizziness and feeling terribly lethargy.

I was having bad cough and fever before. The cough is still on, but better, while the fever has gone.

how long the side effect of vi-q-tuss will last?

According to one source I found, there was no mention of rashes, but the stomach upset, nausea and dizziness were supposed to go away after a few days.

This same source also said to consult your doctor immediately if you develop a skin rash. I'm not sure if this is a site that sells things but I will post the link anyway as it is the source I have cited and it is only fair.
http://www.destinationrx.com/learningcenter/druginfo/info.asp?prodid=00603185358&name=VI%2DQ%2DTUSS

If you develop a rash from a medication, go see your doctor. It could herald a potentially dangerous allergic reaction, especially if the drug is not topical.