It would be better to write 'I am somewhat pessimistic about next year's economy.' OR 'I am somewhat pessimistic about the economy next year.'

Without an article (e.g., "the") the sentences are good but do not mean what you think they mean. "Economy" has a meaning which is close to "frugality" or "thriftiness". Without the article, that is the meaning that would be inferred.

Both sentences are understandable, but they are not fully natural. Here’s how they can be improved: 1. "I am somewhat pessimistic about the economy next year."
This sentence is natural after adding "the" before "economy" and removing "in" before "next year." 2. "I am somehow pessimistic about the economy next year."
This is grammatically correct but less natural because "somehow" implies uncertainty about the reason, which might not be what you mean. If you’re expressing a moderate level of pessimism, "somewhat" is more appropriate. Recommendation: Use the first sentence:
"I am somewhat pessimistic about the economy next year."
It’s clearer and sounds more natural.

Yes, both are ok, but you need an article for ‘economy’: the economy. And also it would be better to say ‘during’ the next year (not ‘in’).