The rules are right here:
https://www.united.com/web/en-US/con...ageCredit.aspx
If the account was open at the time of flight, you can request up to 12 months in the past.
New MileagePlus® members may request mileage credit for flights completed on United, United Express, Continental and COPA up to 31 days prior to their enrollment date. For a $25 USD fee, new members may request mileage credit for flights beyond 31 days and up to six months prior to their enrollment date. Contact the MileagePlus Service Center to process this request.
I'm not sure *A partner flights are allowed prior to the enrollment date.