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This is the description of essence reaver:

+65 Attack Damage +20% Critical Strike Chance

UNIQUE Passive: +10% Cooldown Reduction.
UNIQUE Passive: Gain increasingly more Cooldown Reduction from Critical Strike Chance provided by other sources (maximum +20% additional Cooldown Reduction at 30% Critical Strike Chance).
UNIQUE Passive: Critical strikes restore 3% of your maximum Mana pool.

I wanted to know, if the critical chance of the item itself (20%) is getting involved here as well. When I for instance have 10% critical chance through runes and would buy essence reaver, would I gain 30% cdr instantly?

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No my friend it does not, you can read it here

you can see that they specify

"Gain additional Cooldown Reduction depending on Critical Strike chance from other sources."

EDIT: Sometimes all we need is to read things carefully. :)

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