yes 1.8 k is ok. The resistance has to be low enough that pin 2 can trip the lower threshold. capacitor filters out any funny glitches during reset that may cause another trigger. if you get to hundreds of microfarads then the filter will be slow enough the human senses can tell if its triggering during reset or later.
If you have a N FET or NPN transistor, use it instead of inverter. 7404 has "totem pole" output, normally a reset line wants an "open collector" (or "open drain" in case of FET). Just connect emitter/source to ground, drain/collector to reset, and the base/gate to 555 pin 3 thru a resistor of about 1k to 10k. It will invert, and it will never try to pull up, only pull down.