Misconception: You need a GFCI out in the weather to run your Christmas lights (or whatever.)
More nuanced view from experience: You need GFCI protection for that outlet. You may already have GFCI protection for that outlet that you don't know about. If you spend $3-5 on a plug-in outlet tester with GFCI test button, you can find out if you have GFCI protection on that outlet by plugging it in and pushing the button. If the power turns off, then you get to go looking for where the GFCI protecting it is. They live a lot longer if they are inside the house, out of the weather, with the outside circuit attached to their "Load" terminals.
Misconception: outside outlet boxes must be sealed! Reality: preventing bulk water ingress is good, but all outside conduits are defined as wet locations, and "perfectly sealing" outside boxes tends to mean that condensation does not drain out, rather than that they are dry.
If there's no cable clamp (which might well be sticking out into the lamp-post and require looking carefully from inside the box to notice) and the back of the box is not connected to conduit, that's an issue, but "sealing" is not really an issue, in general. You can stuff conduit openings with "duct seal" to reduce air and water movement, as well as (small) wildlife. That can reduce condensation.
As for mounting a box to a lamp-post, attach it with a couple of screws, drill a hole for the wire access, and either run conduit up to that hole inside the lamppost to run wires in, or use a cable clamp in the back of the box and run cable inside the lamp-post to a junction box with another cable clamp where you transition to wires in conduit, if you have conduit. Many lampposts just have direct buried cables run to them - if they are buried at a sufficient depth and are rated for wet locations and direct burial that meets code, though experience says that running wires in conduit has far less problems in the long run. Edit: actually, the lamp-post itself may qualify as a "conduit" of sorts, so you may not need a conduit inside it to run wires.
However, you have a more extensive problem here - if the cable sheath isn't gray, your cable really is "Romex" or NM/B, and it's not suitable for wet locations - which all exterior conduits are defined to be. The "Romex-like" cable that is suitable for wet locations is UF, and has a gray sheath. So all that cable needs to be replaced back to a point that is inside and dry (if you actually have conduit, you can replace it with THWN wires, rather than cables. Wires are a LOT easier to pull in conduit.)
