essen

Provided you take an RE or RB train you don't need a reservation - in fact you can't make reservations on these trains. On ICE or IC trains reservations are possible - for the short distance between Cologne and Essen however RE and RB trains are perfectly acceptable and frequent (not to mention cheaper than ICE and IC).

As stevegerms said, you could easily take a regional train (runs 1/hour, takes 1hour), they're in fact nearly as fast as an IC or ICE on this short stretch, though they might be slightly more crowded at peak hours - but beware of taking the suburban "S6", as it will literally take ages taking you to Essen. Take a look at the timetables to figure out what will suit you best.

In case you take an IC/ICE, there'll be a display above each seat telling you whether or for what part of the train's journey it is reserved. There could even be a sign saying something like "possibly reserved": A small number of seats is retained for last-minute reservations, and as these bookings cannot be forwarded once the train is on the move, no exact indication can be given. You're free to sit on these, but should be prepared that seatholders might show up and claim that place (they'll be able to produce a printed reciept to proove they're right).

You could reach your hotel in Essen by public transport, too (this would even be included in the ticket price in case you took a RE-regional train ('NRW-Tarif' will be printed on the ticket)), but you would need to change from underground to bus for the last bit. No idea about taxi prices, sorry, but it isn't that far (roughly 5km).