"I am busy these days."
For some reason that doesn't sound very natural. "These days" is okay, but it sounds a little broad compared to the Chinese.
"This is a busy time for me" sounds okay. Otherwise, I'd want the period of time to be specified, e.g. "I'm very busy for the next couple of weeks" etc.
You're assuming that the speaker of the Chinese sentence is talking about recent days/weeks, which may be a valid interpretation of the sentence without context. To me, it actually sounds like the speaker is either responding to a question about his availability over a particular time or is pointing to a period of time on a calendar. Without this context, the Chinese sentence sounds unnatural to me.
In the context of referring to a future period, I think your suggestion would sound a little better with "that" instead of "this", and with the future tense: "I'll be very busy during that (period of) time". Same for a past period, but with the past tense.
Those would be my personal preferences, though I think "this" could still work in either scenario, and I think the present tense could also be used to refer to the near future.
As for the Chinese, I think it could refer to the past, present, or future (though for the past and future we might choose to use "那" instead of "这"), so there are a number of translation avenues. This is a sentence that's apparently pretty highly dependent on context indeed.