Will the James Webb Space Telescope be able to capture something similar to, or better than, the Hubble Ultra Deep Field? Will the James Webb Space Telescope (JWST) be able to capture, or ever be used for, anything similar to the Hubble Ultra Deep Field (HUDF)?
 A: James Webb and Hubble use two different technologies. In reality, James Webb is more of an updated Spitzer telescope, with the range being somewhere between it and Hubble. The reported range of James Webb will be in the 1-25 micron range.
Detectors just simply aren't that great at the higher end of that range, at least compared to visible sensors. Even the best instruments don't work as well as sensors from 15-20 years ago visible. Still, technology has considerably improved recently in this regard. 
Still, the size of James Webb is a three times increase in diameter, which is a nine times increase in collecting area, over Hubble.
My guess is, anything in the Near Infrared (NIR) will be better in its sensitivity than the Hubble Ultra Deep Field, but anything in the Mid Infrared (MIR) will be less sensitive. Still, I can't imagine that they won't try do do something like the Hubble Ultra Deep Field, as its objective, copied from the web site, is:

It will study every phase in the history of our Universe, ranging from
  the first luminous glows after the Big Bang, to the formation of solar
  systems capable of supporting life on planets like Earth, to the
  evolution of our own Solar System.

A: Pushing beyond the Hubble's deep field images is one of the four primary science objectives listed on the JWST site.  The reason we need an infrared telescope to do this is that the earliest galaxies in the universe are being redshifted beyond the limits of the HST's sensors.  IT might not be able to create images that are equally sharp due to the longer wave lengths; but there's simply much more IR light available from that time period than there is in visible wavelengths.

To find the first galaxies, Webb will make ultra-deep near-infrared surveys of the Universe, and follow up with low-resolution spectroscopy and mid-infrared photometry (the measurement of the intensity of an astronomical object's electromagnetic radiation). To study reionization, high resolution near-infrared spectroscopy will be needed.

A: The ability to see deep into the field like this is characterised by the detector's Noise Equivalent Power (NEP). This is a measure of how faint a signal can be retrieved from the 'noise' present in the detector. This noise represents several effects that inevitably create a false signal.
I can't find any numbers regarding HST vs JWST, but I would suggest that a factor of $10^4$ is possible with the technological advance between the two telescopes.
Webb Science: The Assembly of Galaxies describes the deep galactic science that would be carried out by the near infrared camera taking a 'UDF' image...
