The fare was 4.10 dhs and the bus goes from the IBN Battuta metro station to the new DWC airport (or vice versa) with almost no stops inbetween. This F55 bus trip took about 40 minutes and that was in fairly heavy traffic on a Thursday afternoon. The bus is limited how fast it can go on the motorway, so that time-frame will probably be fairly constant in the 30-45 minute range. The schedule and route can be read about here.
You will need a Nol card as Dubai buses no longer accept cash payment onboard. This can be obtained at the metro station, but for those flying in, I'm not sure if they sell them at the new airport or not (although they should). The Silver Nol Card costs 20 dhs, with 14 dhs credit included, and is valid for 5 years. You can recharge or add credit at machines at all metro stations and some other areas such as the large Ghubaiba Bus Station near Diera.
On returning to DWC, I got a taxi to IBN Battuta, which took about 20 minutes and costed me 60 dhs. Since there is only 1 bus per hour you might be better off taking a taxi to the Jebel Ali metro station. That's the first station on the Red metro line, which goes all the way across the city past the other airport. If I did that, it would have probably cost around 40 dhs instead. The fare into Dubai's city centre could be 200 dhs or more depending on traffic so sticking with the metro is a good idea. Once at the metro station, you could then buy a Nol card and get basically anywhere in the city for less than 10dhs; be aware the metro service stops around midnight if you are arriving on a late flight.
I wrote another post on both the Dubai airports which can be read here.