Thermistors will work in just about any temperature until it's so hot they melt.
However you may want to change your pull-up resistor to match the typical NTC resistance for your temperature range. You get most accuracy if your NTC resistance is close to that of your pull-up resistor. A regular 10k resistor with 10k nominal NTC will work fuine, but as you want to measure 1-5°C mostly, a 30k resistor (or a value near it, like 33k) will give better accuracy in that range.