Is it possible to be Arminian and believe in personal predestination to salvation? (as opposed to corporate election)

No, the Arminian would not say that the saved are added to the body of corporately predestined people - just that they are added to the body of saved people. The Arminian does not believe in the predestined. They are added to the corporate body of people (who are not predestined) after They elect to become saved.

This fundamentally comes down to what it means to be "elect". To the Calvinist, this can mean that God predestined or forordained your salvation. To the Arminian however, the "Elect" are just those to which the Abrahamic Covenenat was given.

Paul then extends this into a "New Covenant" in passages like  Hebrews 8:6–13 or 1 Corinthians 11:25, 2 Corinthians 3:6, and Hebrews 9. But this is still a cause which is joined by those with free will who choose to join the cause of the Gospel (The Arminian meaning of "elect"). Not something which is predetermined.

But this is probably the wrong place for this discussion and I hope it is transferred to the Christianity SE by a Mod. But no, it would not be possible to be Arminian and believe in "Personal Predestination" because the Arminian rejects predestination - instead believing that God is continually engaging in creation. This work is not yet entirely complete. The future is not yet forged, but the present is forged by the grace of God and humankind has been invited to engage in the acts of creation - we create art, we create life when conception happens, we were invited in Genesis to name creatures (a creative act in antiquity) and so forth. So creation continues to happen now and nothing can be "preordained" in the strictest sense - only influenced by God who omnisciently sees all creation as it happens and is created.

TL;DR - yes, but not at all in a Calvinistic sense.

I consider myself more or less Arminian, and do affirm individual predestination, but to attempt to give a more objective answer, let's look at the 5 points often used to summarize Arminianism (paraphrased from this article as I generally do not find Calvinist sites accurately represent the details of their opponent's positions, but I couldn't find an Arminian source in my 30 seconds of Googling):

1 - Free Will

Possibly the most misrepresented and misdefined doctrine, Arminians generally hold to libertarian free will. This can generally be defined as the ability to do otherwise; more specifically, for any action a man does, nothing caused or necessarily brought about that action such that he could not have done something else, or at least refrained from doing the thing he did. "Nothing" includes that man's own desires, excluding a compatibilistic determinism.

This point pretty clearly rejects the idea that God is in any way puppeteering us, or even that our desires determine what we do.

2 - Conditional Election

This essentially means that the election of an individual is conditioned on their accepting of the Gospel, which God foreknew from eternity past. In other words, a person is elect simply due to the fact that they will accept the Gospel, which God has always known. Alternatively, God only chooses to bring to faith those He knows will respond positively of their own accord.

Not all Arminians hold to this view; some would instead affirm corporate election, whereby a man becomes elect upon belief; the elect are those who believe, not those whom God has decided will believe.

This pretty firmly establishes Man's choice/role in his own salvation; he doesn't contribute to it, but accepts it of his own accord.

3 - Universal/General Atonement

Christ's death was sufficient to save everyone, but is only effective/applied to those who believe.

This precludes predestination in the sense that it is not the case that Christ only died for some people, who will certainly then be saved, but salvation is instead genuinely available to all.

4 - Resistible Grace

This directly contradicts a key point of Calvinism, stating that God's grace is resistible; one could receive a genuine call to salvation and have just as much opportunity/grace as another, but choose to reject it while the other does not.

This again emphasizes that it is genuinely a choice to accept Christ, not something that is predetermined in which we have no other option.

5 - Falling from Grace

This essentially means that it is possible to forfeit one's salvation, not simply by sinning too much but by deliberately turning away and rejecting God.

This pretty clearly rejects Calvinistic predestination, as even once saved a man still has the choice to leave.

So after all that, you might wonder how I could still affirm individual predestination of any kind, and the short answer is Molinism, which I won't get into here but have described in other answers on this SE.