It all goes back to what are prerequisites to you?
The effective ability to cast 2nd level spells means: I can haz cast 2nd level spell form wand!
Then yes. Versatile Spellcaster certainly is the right thing for you. Except wands do it better.
If you want the ability to cast 2nd level spells.
Then yes. Versatile Spellcaster certainly is the right thing for you. Except Spell Completion items do this too.
If you want the ability to cast 2nd level spells, as in a spellcasting trait with the ability to cast 2nd level spells.
No, you don't obtain that until later on.
Also, regardless, it's TO to say it's any spell as in Sorcerer, or Cleric, or Wizard, or Druid. Spellcasting speaks in a possessive way, when it says a spell you know (any is in the example which is in an only-sorcerer build btw) it's only speaking of the same type. Exactly like how a Specialist Wizard can't ever learn spells from a certain school, or Cleric can sacrifice a prepared spell they have, etc. These traits do not carry over into other classes, if you combine to 3rd level Sorcerer slots then you'd be allowed to cast a 4th level Sorcerer spell, not Druid or anything else you think you have.
P.S. Oh I almost forgot. To cast a 4th level spell you still need a CL of 8 or higher, Versatile Spellcaster does not overwrite the base rules on minimum caster level.