A bit of good old Google-fu reveals this blog post by Emily Lakdawalla for when New Horizon's LORRI imagery of Pluto is expected to be of better resolution than that of Hubble Space Telescope's Wide Field Camera 3 (WFC3) (relevant excerpts only, no copyright infringement intended):
So New Horizons' vision is is about 25 times less sharp than Hubble's.
Therefore, New Horizons will have to be about 25 times closer to Pluto
than Earth is before its images will begin to improve on Hubble's.
Right now it's "only" 4 times closer to Pluto than Earth is. It still
has a long way to go!
Pluto will appear 3 pixels across to LORRI when New Horizons is about
160 million kilometers away. How did I get that number? Pluto is 2400
kilometers across, so 3 pixels means each pixel is 800 kilometers.
Divide that by 5 microradians -- which is the same as dividing it by 5
and then tacking the word "million" on the end -- and you get 160
Her calculations regarding the date when that's supposed to happen (1st of January, 2015) are a bit off from dates quoted in Wikipedia article on New Horizons' key mission dates, though:
Feb 2015 Observations of Pluto begin
New Horizons is now close enough to Pluto for the main science mission to begin.
5 May 2015 Better than Hubble
Images exceed best Hubble Space Telescope resolution.
This discrepancy suggests there were quite some approximations involved in the calculation of the date and uncertainty on which of the two calculations is more correct, or which of the Hubble Space Telescope's cameras is meant in the Wiki page adds to confusion, but math behind the calculations of actual distance when this will happen (as per your question) appear correct (although the numbers used mightn't be).
This obviously called for a peer review, and Derek Szymanski was more exact in the comments to the mentioned blog, and uses more precise resolution numbers for Hubble Space Telescope's Wide Field Camera 3 (WFC3) (~ 2.68 pixels wide photographs of Pluto), coming up with a distance of roughly 180 Million kilometers.
Do note though these calculations are for the Hubble Space Telescope's Wide Field Camera 3 (WFC3), and not the ESA Faint Object Camera, which has approximately 5.3x finer resolution (by my calculations)* and has produced this photograph:
ESA Faint Object Camera's views of Pluto and computed interpolations (Credit: ESA)
According to NASA's page describing this photograph, each pixel is over 150 km across. Taking diameter of Pluto into account (2390 km), this translates into resolution of roughly 16 x 16 pixels, and the photographs appears to confirm that.
As for the other part of your question, what sort of work would the New Horizons team be able to do at that time, this is pretty much when the LORRI images become more usable than HST's own observations, but observations of Pluto by LORRI begin months before that (Feb. 2015). Wiki provides these details on Pluto flyby mission objectives:
Primary objectives (required)
- Characterize the global geology and morphology of Pluto and Charon
- Map chemical compositions of Pluto and Charon surfaces
- Characterize the neutral (non-ionized) atmosphere of Pluto and its escape rate
Loss of any of these objectives will constitute a failure
of the mission.
Secondary objectives (expected)
- Characterize the time variability of Pluto's surface and atmosphere
- Image select Pluto and Charon areas in stereo
- Map the terminators (day/night border) of Pluto and Charon with high resolution
- Map the chemical compositions of select Pluto and Charon areas with high resolution
- Characterize Pluto's ionosphere, and its interaction with the solar wind
- Search for neutral species such as H2, hydrocarbons, HCN and other nitriles in the atmosphere
- Search for any Charon atmosphere
- Determine bolometric Bond albedos for Pluto and Charon
- Map surface temperatures of Pluto and Charon
It is expected, but not demanded, that most of these objectives will be met.
Tertiary objectives (desired)
- Characterize the energetic particle environment at Pluto and Charon
- Refine bulk parameters (radii, masses) and orbits of Pluto and Charon
- Search for additional moons, and any rings
More can be read here.
* Correction: My calculations on the resolution of WFC3 vs. FOC were fairly accurate, considering they were based on approximate data and visual observations alone, but I've nevertheless found more precise resolution difference (slightly in favor of FOC), which is in the order of 5.55x, with WFC3's resolution of 0.04 arcsecond pixels as mentioned before vs. FOC's 0.0072 arcsecond pixels.