Learn Russian: Завтра я планирую познакомить вас с одним инвестором. - Tomorrow I plan to introduce you to a certain investor.

In this context, одним doesn't act as a number. Instead, it acts as an indefinite specific pronoun meaning "a certain". The speaker has a specific person in mind but is choosing not to name them or give more details right now. It functions quite similarly to the English indefinite article "a" or "an" in certain contexts.

Could I use каким-то instead of одним here? How would the meaning change?

You could, but the nuance is very different. Using один in this indefinite sense implies the speaker knows exactly who the investor is, even if they aren't naming them yet. Using каким-то would imply that the speaker doesn't really know who the investor is, giving the meaning "some investor or other".

How does the verb познакомить dictate the cases used for вас and инвестором in this sentence?

The verb познакомить (to introduce or acquaint) requires the accusative case for the person being introduced, which gives us вас. The person they are being introduced to requires the preposition с (with) followed by the instrumental case, which gives us с одним инвестором.

Why do we use the perfective verb познакомить after планирую instead of an imperfective verb?