How long should I wait to put in an OCS packet after arriving to my first duty station?

Just like what’s already been mentioned, you can pretty much apply as soon as you get to your unit after AIT. I enlisted with a degree and waited a little bit under a year after getting to my unit to apply for OCS. It really all depends on your command and their recommendation of you to attend OCS. You’re company, battalion, and brigade commanders will all have to sign atleast one document each detailing they support/recommend you for OCS. Get to your unit and prove your worth right away! Don’t hesitate to ask your command what they expect of you and what it’ll take for them to recommend you. The application process will be driven completely by you. It’s a lengthy process but doable. Let me know if you have any questions!

As soon as you're eligible IAW the guidance in the most recent MILPER.

As far as I know, there isn't any regulation that stops you from applying the day after you report. On the practical side, getting into OCS will generally require a recommendation from your commander, so how long he/she feels that you need to be in the company before that happens is largely their call. If you are interested in OCS, when you hit your first Duty Station, talk to your Chain of Command about it, which should include your Company Commander. You have plenty of time.