The current deal is as low as 399 dhs roundtrip inclusive which is around $110 USD! While this promotion will be over soon, you can often find some great deals through Air Arabia and I'd definitely fly with them again (and probably will soon). Some of the main destinations they fly to are Turkey, Sri Lanka, India, Nepal, Kenya, Oman, Jordan and Morocco. They’ve even added flights to Switzerland, Spain, France, Netherlands, Ukraine and Russia, so Air Arabia is a great way to get from Europe to the Middle East or vice versa. A complete list of destinations can be on their website here.
The airport has bus service to Sharjah city centre. Public bus route 111 (costs 10 dhs) runs hourly from Sharjah's airport via Al Qusais Bus station onto the Metro's Rashidiya station in Dubai which is very close to Dubai Airport (DXB) Terminal 3. A taxi between the airports or to Dubai's city centre should be around 100 dhs depending on traffic.
Keep on travelling