Is this correct?
The person with whom I'm doing the project should be here soon.
If it is, is "with" always a dative preposition? (like "mit" in German)
When "who" is the object of the preposition, as in this case, it becomes "whom"; granted, this is archaic and often ignored in informal conversation. You'll often hear people say things like, "Who should I give this to?" It would be correct to say "Whom should I give this to?" or even (if you're really fussy) "To whom should I give this?" But almost no one bothers with that these days. Note that reversing the word order makes the incorrect grammar stand out: "I should give this to who?" That's because there is now a direct apposition with the preposition and its object. Most careful speakers will use "to whom" in that context.
You can remember when to use "who/whom" by substituting "he/him" in the sentence. You wouldn't say "I'm doing the project with he," you would say "I'm doing the project with him." So it's obvious that whom is the pronoun you would use here, not who.
A further word about German/English prepositions. In German some prepositions can be dative or accusative, depending on whether they indicate motion or placement towards or up to a location. This not the case (no pun intended) in English. In English, the object of the preposition always takes the "prepositional" case. Note that there are not nearly as many inflectional changes or pronoun substitutions in English as in German. The point is, German is not necessarily useful for analogizing English constructions.
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Here is a really easy way to deal with case and prepositions:
If the the preposition is directly modifying the noun, then the noun is always* accusative/dative. And, since accusative and dative forms both look like whom, you know it should always be whom (if you are using whom at all).
*There is one major exception to the prep. phrase rule: of sometimes takes the genitive case ("friend of his"); however, this never shows up with who (there is no "friend of whose", just "whose friend").
It is worth also noting that whom is falling out of use, and so who could theoretically be used everywhere when speaking in a register that doesn't use whom.
In your case, the person is the object of the sentence, while I'm is the subject. Even though the sentence has a questionable structure, it's correct
This article does a remarkable job explaining when to use who and whom.