Is it natural to say
The pop-up message here does not only carry sufficient information but also is confusing
when I have these two facts:
Also, should I insert "rather" somewhere in my sentence?
An earlier answer has already said this, but I think it bears repeating:
Your current structure does not convey the facts you mention.
The vital thing to understand here is that the words "not only" do not negate the words that come after them. When you say or write, "not only X," you affirm that X is true, and you imply that there are other things that can be said on the same topic. "Not only X but also Y" is a special way of saying "X and Y." This phrasing might be used if X is something good and Y is better, or if (as in this case) X is something bad and Y is worse. It can also emphasize both X and Y.
Since "not only X but Y" means "X and Y", one way to put together your facts 1 and 2 is,
Not only does the pop-up message not carry sufficient information, but also it is confusing.
This now has the correct meaning, but its style can be improved. First, rather than "not ... sufficient", we can use "insufficient":
Not only does the pop-up message carry insufficient information, but also it is confusing.
Because the words "not only" imply that there is more to follow, the word "also" in "not only ... but also" is redundant:
Not only does the pop-up message carry insufficient information, but it is confusing.
This has the disadvantage that the word "but" often conjoins two things that contrast with each other, whereas here we have one thing that is like the other but more stronger. It would be better to omit the word "but", which is not needed in this particular sentence:
Not only does the pop-up message carry insufficient information, it is confusing.
In this sentence, the comma is sufficient to separate the two facts that you want to state. The construction "not only ... but" is an idiom that is useful when the two elements following "not only" require a word to separate them, usually because they are not complete statements by themselves:
The pop-up message is not only uninformative but confusing.
A note on the possible use of the word "rather" in any of these sentences: "rather" has several possible meanings. From http://dictionary.cambridge.org/us/dictionary/english/rather:
in preference to or as a preference
more accurately; more exactly
to a noticeable degree; somewhat
The second sense often occurs in the construction "not ... but rather". For example, you might say
The popup is not an informative message but rather a source of confusion.
If you write
The popup is rather confusing,
a listener will likely take "rather" in its third sense; the sentence is understood to mean that the popup is somewhat confusing. This meaning is likely to be conveyed even if the word "rather" appears within a sentence where two things are being compared or contrasted.
Your current structure does not convey the facts you mention. In its current state, it is saying that carrying sufficient information is not the only thing it does, when it should say that it doesn't carry sufficient information (or carries insufficient information). The wording I would choose is:
Not only does this pop-up message carry insufficient information, but it is also confusing.
I would do it this way, using not only...as well:
Not only does the popup carry insufficient information, it is confusing as well.
P.S. I see the OP uses inversion, but for other passersby, note the inversion: "Not only does the popup..."
It is possible to combine these two complete ideas in one sentence using not only.. but also..
The pop-up message not only carries insufficient information but, worse, it is also rather confusing.
The not only.. but also.. construction is challenging to use and contributes no more to the meaning than the much simpler furthermore. As you want to emphasise the second point, it might be better to omit it:
The pop-up message carries insufficient information and, worse, it is rather confusing.
I would say: "The pop-up message doesn't carry enough information and is quite confusing."
