Welcome to the Blades community! We’re always in need of fresh blood.
(Especially since some have demons and vampires to feed)
Regarding when a Ghost can unlock the Possess skill, the manual only states that “When you transfer your character to the Ghost playbook (…) you may also choose one more Ghost trait”. My interpretation of this wording is that, when you either start as a Ghost or transfer from another playbook into a Ghost, you automatically start with Ghost Form and another Trait of your choosing. Since the manual also adds that “you may not choose Possess as a starting trait”, this simply means that you cannot choose to have Ghost Form + Possess as your starting traits.
However, there seem to be no further requirements or restrictions for unlocking Possess, which means that as soon as the Ghost fills their XP tracker for the first time, they will already be able to unlock the Possess skill, as far as I can tell.
Other than that, Ghosts have need instead of vices, suffer drain instead of stress and gloom instead of trauma. That’s all just flavourful semantics, really. Nothing changes from a mechanical perspective. There is nothing specifically stating that, during Downtime, the Ghost characters lose access to anything, much like the Hull and the Vampire.
Does it make sense for a Ghost to have long-term projects? My opinion is: absolutely! This PC will very likely be one of the selective ghosts in Doskvol who has somehow kept their sanity. This means that, although they may be dead, they still (mostly) think like a living person. And all living people have motivations and goals, which lead to projects.
Is it a bit weird for a Ghost to do Downtime activities such as Train? Maybe. But the book doesn’t say anything against it. And Weird is sort of part of this game if we’re being honest. Also, it’s always up to the group to decide how they want to approach this. Will is just be a mechanic roll and glossed over? Or will you, as the GM, actually ask the player how they are training? Possessing a victim to train through them is the first thing that comes to mind, but the player may have other - more original - thoughts on the matter. You know what they say: play to find out!