How would a physicist measure temperature of molten metals in 1850-1920s? What equipment would be used?
The Aethrioscope (see Wiki page with this name) was invented in 1818 by Sir John Leslie and the basic idea for a pyrometer (see Wiki pahe with this name) was conceived in the late 1700s by Josiah Wedgewood. These were calibrated by comparing observed colour with that of hot metals / clays (as appropriate) of known temperature. The idea was to heat a small amount of the substance under test and estimate its temperature through thermal expansion, then to use this to calibrate the pyrometer / aethrioscope. The diffraction grating and the spectroscope (see also the wiki page for spectroscope) were invented by Joseph von Fraunhofer in the very early 1800s (1814 for the spectroscope and 1821 for the grating). Samuel Pierpont Langley invented the bolometer in 1878](http://en.wikipedia.org/wiki/Bolometer).
So, in short, pretty much all the standard techniques we use today to measure the temperature of hot metals were available by 1850, the only difference is we have edvanced in our knowledge of what gives rise to the spectrums and thus we have more theoretically sound calibration procedures.
Indeed it was the development of all these instruments up to the time period that you cite that let Planck to do his groundbreacking work on the quantised theory for electromagnetic radiation in black body radiators, not, as is often thought, the other way around.
One main advance that happenned just after your cited period was the development of the ratio pyrometer, which used the intensity at many colours to make a more robust (less affected by varying surface emissivity) measurement.
In summary: "the same techniques as now, only with somewhat less accuracy".