With a thermometer, I guess.. the oldest record is this one:
https://en.wikipedia.org/wiki/Central_England_temperature
Ocean temperatures were also measured during the research voyages around that time.
However, it is also possible to deduce temperatures from closed-off water bodies such as ground water or water inclusions in rock (such as stalagmites).
While radioactive decay can give you the time since last exposure to open air (different elements have to be used depending on the expected age range), ratios of other things such as noble gases can give you the temperature at that exposure. (Different gases have different solubility curves in dependence of temperature.)
Ground water bodies can be over 1000 years old, so 1880 is definitely within the timeframe. At the same time, the techniques are also used on ice cores. The time resolution can be quite different between the techniques, though.
Similar techniques can also be used for ocean temperature, I presume.
Now, we also use thermometers. There are distributed networks of weather stations. You could argue that newer thermometers have a higher accuracy, but at the same time it is very hard to create an environment where you measure air temperature only, and are not influenced by parameters such as irradiation. Thus, as the page mentions, normally only deviations are given.
Plus, of course, satellites give a much better spatial resolution. (Have fun with atmospheric layers then.. with of course different temperatures.
https://en.wikipedia.org/?title=Satellite_temperature_measur...)
Here is a very detailed publication on the topic of isotopes in water, just in case you are bored:
http://www-naweb.iaea.org/napc/ih/IHS_resources_publication_...