WeatherData
can be told to give you all the weather stations it knows about, but there's more than one type of weather station (and duplicates!):
allWeatherStations = WeatherData[];
fullformStations = FullForm /@ allWeatherStations;
Tally[StringLength /@ fullformStations[[All, 1, 2]]]
(*{{4, 6919}, {5, 10306}, {8, 4697}}*)
Looking at the documentation, I find the "WMO*" stations seem to be the interesting, and reliable ones. I extract these and find their corrdinates:
wmoWeatherStations = Cases[fullformStations[[All, 1, 2]], a_ /; StringLength[a] == 8];
wmoStationsPositions =
GeoPosition[WeatherData[#, "Coordinates"]] & /@ wmoWeatherStations;
Using GeoWithinQ
I find those stations within Mongolia and then use Pick
to include these in GeoGraphics
wmoMongolia = GeoWithinQ[Entity["Country", "Mongolia"], wmoStationsPositions];
GeoGraphics[GeoMarker[Pick[wmoStationsPositions, wmoMongolia]],
GeoRange -> Entity["Country", "Mongolia"]]

CountryData["Mongolia", "Shape"]
$\endgroup$]
at the end. Also note its good Mathematica practise NOT to use Capitalised variable names as you may inadvertently collide with a named Mathematica function. $\endgroup$