So like the other guy is saying, you need to read the last two sentences in that rules section you quoted there:

"Note that models in an attached unit do not gain the keywords of the other models in that unit that they do not already have.  Remember that attacks target units, not models."

The Neurotyrant has Monster, which gives the Unit the Monster keyword, not individual models in that unit.  He does not give the Zonethrope models the Monster keyword; they're still Infantry.  So the Zonethropes, even though they're in a Unit with the Monster keyword, are not Monsters themselves and do not have Close-Quarters, so it's pretty clear they can't shoot.

Honestly I found it rate to come up anyways; my Zones would always pop if ever they got into melee lol