If you can hire a car, that would be your best bet. It's about 60-90 minutes away depending on traffic/how fast you drive.

I reckon you'd be able to agree £50 with a taxi driver.

If you know when you need to go to DOH, just book MAN-DXB-DOH-DXB-MAN with a stopover in DXB. This will generally be the same price, plus maybe a bit more for taxes.

Cheers,
Rick