The LED strip is only 1.8 watts and your proposed supply can supply 12 watts (12 volts out x 1 ampere capabiliy = 12 watts). Or if you prefer, the LED supply current for 1.8 watts is only 0.15 amps (1.8 watts / 12 volts = 150 milliamps). So the adaptor isn't overloaded.
From what's on-line, there is not enough information to know if the LED strip has current limiting resistors built in or not. I would expect that is part of the design.
Update: looking at the detail photos of the LED strip I can see surface mount resistors between the LEDs. You shouldn't need any additional resistor unless you want to dim the lights down some more.
Eheim Pimp #459 - 2213, 2234
SunSun Pimp #47 - (Perfect) HW302