Once you set a price for a mail service (and certainly if you already paid), the non-Jewish worker can take a letter from you even right before Shabbos. (ShA1)
A case that this would be prohibited is when you tell him “go on Shabbos”. Also, do not tell him “make sure it arrives on Sunday” if you know it is impossible without taking the parcel on Shabbos as well. (MB4)
You may not allow a gentile to pick up a parcel at your house on Shabbos. (MB2)
You may not put a letter in the post box on Shabbos even if there is an eruv. In case of great financial loss, you can tell a non-Jew to put it in the post box. (MB17)
Do not instruct a non-Jew even on erev Shabbos to put a letter into the post box on Shabbos if there is no financial loss. (MB17)