F5 to Run Shell script on remote host in certain event

so basically the idea is to generate syslog trap on F5 and use that to trigger a script ?

yes. you can execute script using syslog-ng (Kevin's suggestion), alertd or icall.

this is alertd.

sol14397: Executing a command or custom script based on a syslog message

and this is icall.

iCall - All New Event-Based Automation System by Jason Rahm

is it an iRule ?

no, it is syslog-ng configuration.

to run script remotely, you may check if plink is usable.

Plink