First of all, the longitude will be very difficult to pin down, since it’ll mostly be moving along the horizon, not up or down.
But assuming you do pin it down, no, it won’t be at the same longitude every year. That would only happen if the year were an exact integer number of days. Since it’s about a quarter-day longer than an integer, the sunrise point will move almost 90 degrees each year.
And it won’t peep up briefly before going back down, either, at least not if you’re right at the pole. Right at the pole, there is only one sunrise and one sunset per year: It spirals up gradually for three months, then spirals back down gradually for three months, and then is dark for six (during which it’s still spiraling, just below the horizon where you can’t see it).
This all assumes that the horizon is perfectly uniform and level. If it’s not, then the first sunrise will probably always be at the lowest point on the horizon, or close to it (though the date of the sunrise will vary).