I am new to lead guitar and I want to know how long it would take for me to play a song like that.
I have played the keyboard for years now and I also compose my own music using Logic Pro.
I began learning how to play Acoustic Guitar on February of this year and I practiced every day for around an hour.
Right now I can play every major & minor chord, barre chords, most strum patterns, basic fingerstyle patterns, and I can play along with a song now.
I really want to learn how to play lead and I am willing to practice every day. Where should I start?
Should I practice scales? Should I practice the specific song that I want to learn first? How long would it take with daily practice to play the song I linked?
it's virtually impossible to say because it depends on how you practise as well as time (plus how good you are already).

speed mechanics for lead guitar (book) by troy stetina is pretty good

other than that just learning songs whose solos you like is probably the thing
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Yeah, there's way too many variables to tell you how long that'll take with any degree of reliability. Just keep playing and practicing, and eventually you'll get there. That's really all there is to it.
Really not that long if you plan on playing that sloppy.
You should start by finding patterns that you enjoy and play them with a metronome.
Gradually speed up the bpm and once you hit your max, with no mistakes, start back at the original bpm you began with.

*Make sure you're practicing something that is musical.*
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Everyone is completely different, you might be able to do it in a month or it might take 5 years but i'd say it'd take most player at least a couple years to play that and play it well and believe me you absolutely don't know all the chords, learn songs first, without the knowledge of songs and solos then scales aren't gonna make a lot of sense and if you want to play that song reasonably soon you'd want to be playing for maybe 3 hours a day
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Like Dave already mentioned, it's an impossible question to answer.

I have this one buddy of mine that has only been playing for 3 years, yet he can blow through Galneryus, Versailles, and Dream Theater songs like they're nothing. On the other hand, I know quite a few people that have been playing for much longer and can only play the basics.

More than anything, I think it entirely depends on what you want to play. If you want to play material similar to Sonata Arctica, then focus on that.