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.  
Source: (486958) 2014 MU69, Wikipedia
Various estimates of 2014 MU69's diameter have been made:
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. 
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.
1 Green, Jim (12 December 2017), New Horizons Explores the Kuiper Belt, 2017 American Geophysical Union (AGU) Fall Meeting in New Orleans: 12–15.
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.
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.