Did they just put a ruler on the photographic plate? Quite some precision must have been required to discover Neptune from its influence on Uranus for example.
Prior to digital imaging then photographic plate negatives were analysed with scanning microdensitometers to produce astrometric catalogues. Many of these catalogues are still in use today, they are valuable sources of early epoch positions that enable proper motion measurements.
For more details you could look at the descriptions of the SuperCosmos project http://ssa.roe.ac.uk// (which is based on Schmidt plates) or the UCAC4 catalogue, http://ad.usno.navy.mil/ucac/readme_u4v5, which uses plates to get positions and proper motions for faint stars.
In the good old days, before even these catalogues existed, astronomy groups would have copies of the whole Schmidt and Palomar sky surveys. You would put the relevant plate on a massive, concrete-based, hydraulic X,Y measuring machine, with a binocular microscope. You would measure sets of Standard stars, get a 6-coefficient fit to convert x and y into RA and Dec,then find your objects on the plate, measure x,y, calculate RA and Dec. Then snap a polaroid to use as a finder chart at the telescope.
I was doing this as late as 1995 before the advent of the Digitised Sky Survey.
Micrometers and blink comparators allowed considerable precision.
Pluto was discovered with the aid of the latter.