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Ultima Thule is thought to be about 20 miles (30 km) across. However, I can't seem to find any information about approach distance or typical image resolution of New Horizons in regards to this second target.

What I am interested in is that, since Pluto is a good deal larger than Ultima Thule, we knew a lot more about it before we got there. There are several technical details known about Pluto but not about Ultima Thule due to its dimness (it is a million times dimmer than Pluto), distance, and the fact that it was only discovered 4 years ago. Presumably knowing these details in respect to Ultima Thule would aid the technical aspects of imaging it.

What meter/pixel resolution could we expect to get from such a small target?

Update 2019-01-01: I've calculated detailed resolution and range values for the encounter's imaging schedule. The chart can be found in this answer (scroll down to Resolution during capture in meters per pixel).

I was gathering data to do the math and came across this:

New Horizons is planned to come within 3,500 km (2,200 mi) of 2014 MU₆₉, three times closer than the spacecraft's earlier encounter with Pluto. Images with a resolution as fine as 30 m (98 ft) to 70 m (230 ft) are expected. [1] [2]

^{Source: (486958) 2014 MU₆₉, Wikipedia}

Various estimates of 2014 MU₆₉'s diameter have been made:

• 
  25–45 km ^{3 (2014)}
  


• 
  30–45 km ^{4 (2014)}
  


• 
  30 km or binary pair of 15 to 20 km each ^{5 (2017)}
  

^{Source: ibid, as cited}

The resolution of the imagery to be collected is discussed in detail in another answer

Note that "Ultima Thule" is currently just a nickname for 2014 MU₆₉. As partially quoted on the above-linked Wikipedia page:

[W]e’re going to give 2014 MU69 [sic] a real name, rather than just the “license plate” designator it has now. The details of how we’ll name it are still being worked out, but NASA announced a few weeks back that it will involve a public naming contest. [6]

Another source⁷ adds:

After the flyby, NASA and the New Horizons team will choose a formal name to submit to the International Astronomical Union, based in part on whether MU69 [sic] is found to be a single body, a binary pair, or perhaps a system of multiple objects.

Citations:

¹ Green, Jim (12 December 2017), New Horizons Explores the Kuiper Belt, 2017 American Geophysical Union (AGU) Fall Meeting in New Orleans: 12–15.

(PDF)

² Johns Hopkins University Applied Physics Laboratory (2017), New Horizons Files Flight Plan for 2019 Flyby, 6 September 2017, web page, retrieved 2018-12-27.

³ Buie, Marc (15 October 2014), New Horizons HST KBO Search Results: Status Report, Space Telescope Science Institute, 23.

(PDF)

⁴ Lakdawalla, Emily (15 October 2014), Finally! New Horizons has a second target, Planetary Society blog, Planetary Society, web page, retrieved 2017-12-27.

⁵ Bill Keeter (3 August 2017), New Horizons' Next Target Just Got a Lot More Interesting, NASA, web page, retrieved 2018-12-27.

⁶ Stern, Alan (28 April 2017), No Sleeping Back on Earth!, NASA, web page, retrieved 2017-12-27.

⁷ Tricia Talbert (13 March 2018), New Horizons Chooses Nickname for 'Ultimate' Flyby Target, NASA, web page, retrieved 2017-12-27.