How long does it take to travel from Dubai to Sharjah?

No, to get from Dubai to Sharjah you just need to cross the road. The fact is that Dubai, Sharjah and Ajman actually form one urban agglomeration and there are no districts between them; one city smoothly flows into another. This is similar, for example, to moving from Moscow to Mytishchi. you just need to cross the Moscow Ring Road.
Therefore, you can also get to Sharjah from Russia not directly to its airport, but through Dubai airport, where there are many more flights from the Russian Federation (but it is not a fact that they are cheaper). The distance from Dubai airport to Sharjah airport is only 30 km (this is even less than the distance from Domodedovo-Vnukovo). And the border of the emirate of Sharjah is only 7 km by car from Dubai airport.
A taxi from Dubai Airport to the center of Sharjah will cost approximately 100 dirhams (2,000 rupees). The journey takes about 40 minutes.