How good is Venomoth against Exeggutor?

Bug Buzz/Bug Bite works well enough from Venomoths and Butterfrees. Parasects are normally my bug of choice though. To be honest Exeggutor is pretty weak to the right attacker - Pidgeots work well too.

Venomoth with the correct move set can be useful against Exeggutors but in my opinion are only useful for training friendly Exeggutors in gyms. I would not recommend investing stardust as the additional CP would decrease the Moth's value for training and in terms of attacking enemy gyms, there's really no better counter than Arcanine which has much easier access to higher CP

Not really, I wouldn't advise using Venomoth given the lowish stats. It is usable, given that you go double super effective into exeggutor, but it is not the best choice as a bug type, mainly because of the poison typing which takes super effective from psychic moves.

If your arcanine is in a gym then I suggest using a Flareon if you want to use a fire counter, it's the next best thing.

Bug types that do the better job include Scyther, Pinsir and Parasect (with X scissor).

good because of double weaknesses, but also bad because it's weak to physic moves

You're better off with a Parasect (maybe even Pinsir and Scyther) that has bug attacks or a fire Pokémon like Arcanine or Flareon. Parasect doesn't receive super effective damage from Exeggutor's psychic attacks and X-Scissor is fast enough to be used between Exeggutor's quick attacks, unlike Bug Buzz which is a fairly slow attack.

For attacking enemy gyms, Arcanine with Fire Fang+Fire Blast or Bite+Fire Blast are probably the best choices.

For training team gyms, Parasect with Bug Bite+XScissor or Fury Cutter+XScissor is best.
Scyther and Pinsir with Fury Cutter+XScissor are good choices too, but rarer so harder to get that moveset (pretty much the only thing Pinsir is good for actually).
Parasect and Scyther both have double resistances to grass type moves, which can help train against Exeggutor with Seed Bomb or Solar Beam.

Venomoth has the right type moves to fight Exeggutor, but is weak to psychic type, so will be hurt a lot by Exeggutor's quick moves, and Bug Buzz has a longer duration so you will be getting hit by at least a few quick moves without dodging.

Venomoth with Confusion+Psychic can be a decent choice for training against grass+poison types and fighting types.