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I want to say:
Studying Japanese is difficult, but fun.
I've come up with:
勉強は日本語むずかし、でも楽しい。
Google Translate seems to like it ok, but is this the correct structure? Seems like it could be a bit backwards, but having "study" at the end didn't work out.
(You forgot an い at the end of むずかしい, but this is not the main point of my answer).
Let's try to analyze your attempt. You say 勉強は hence you are making 勉強 the topic, which kind of makes sense given what you want to express.
However, what follows does not really make much sense. In fact, if you try to make a grammatical analysis of the rest of the sentence, what is 日本語 here? There is no particle nor any other structure to precisely identify its role in the sentence. In other words, your try is grammatically wrong.
What you probably want to do is to connect 日本語 with the 勉強 in order to express "The study of the Japanese language", that can be done using の:
日本語の勉強は 。。。
Now the topic is actually "The study of the Japanese language".
The rest of the sentence is quite OK. Just add むずかしい to describe it as difficult and then add the "but" part. I probably would use けど here rather than でも. It sounds more natural and conversational to me, but I am not a native speaker so let's wait for other comments.
Anyway, putting all together your sentence becomes:
日本語の勉強はむずかしいけど、楽しい（よ）。
(You probably can add the よ in the end for a little emphasis).
Another possibility, though maybe it goes a bit far from your attempt, is to change it this way：
日本語を勉強するのはむずかしい。でも、楽しい（よ）。
Or as above with けど、 instead of 。でも、
I put the other first as I tried to stay close to your original attempt.
Your current answer is grammatically incorrect, although I think someone could figure out what you meant pretty easily.
The thing with 勉強 not wanting to work at the end has to do with it being a noun. This is a word, however, that you can turn into a verb by tacking ～する onto the end. That gives you 勉強する, which means "to study" and is more in the vein you're looking for. We're not done yet though.
Because of the way you've phrased your sentence in English, the closest translation I can think of would be 「日本語を勉強するのは難しい、けど楽しい」. This phrasing is where we hit some new material. V+の translates literally to "the thing of V-ing" and allows you to describe actions as if they were noun clauses. This bit of grammar may be a tad advanced at the moment, but I'm sure you can handle it.
勉強は日本語むずかし、でも楽しい
Would sort of translate as :
As for studying, Japanese is hard but fun.
You might want to try this :
日本語の勉強はむずかしいですが、楽しいです。
Studying Japanese is hard but fun.
Also, using Google translate as a reference is quite... unwise. Be careful.
