How can I monitor the temperature in a hot water heater?
I don't see why not. In fact, all heaters I've seen are sold with a temperature probe on the inside and a display on the outside. I don't know why yours isn't. The problem is, of course, that you need to somehow get the probe on the inside - you must find a way, we don't know the construction of the heater. 

The easiest thing to use (provided you can think of a way to insert the probe) would be a digital roasting thermometer. 



The probe is stainless steel, so it won't rust in the water. It has a flexible connection to the processor/display unit. It has a limited temperature range, but 15°C to 100°C should be covered. The price is in the 20 - 30 Eur range. Maybe you can even put the "alarm when a temperature has been reached" into use, or get one of the radio models - they display the temperature on a second wireless display within a limited range, probably strong enough for another room. I don't know what the drain on the batteries will be during constant use, it might be a drawback. So try to get a mignon powered one instead of button cells.