Be direct, but also casual, and make the invitation about the thing you want to do more than about doing that thing with her.
Platonic, casual, innocent invitations to friendly acquaintances do not mix well with intense thought and planning. Acting awkwardly, nervously, or making a big deal of the invitation or event will cut against the impression you want to give-- that this is a low-key outing to see something you would both enjoy, and that you thought to invite her because you know she would enjoy the museum.
Sufficient awkwardness or other significant social missteps can be off-putting whether there is a potential romantic element or not. I have been invited to do things, in situations in which I am certain I am not being propositioned for dates, but something in the other person's affect seems off to me (they're too intense, too nervous, or they seem like they want this to happen a bit too much, etc.), and if I feel uncomfortable enough I decline.
This kind of situation comes up often when one person does have romantic intent and doesn't know how to express it well, but that isn't the only time, and it doesn't have to be the romantic angle that causes the problem. I think that making sure your invitation is viewed as definitely 100% not romantic can backfire.
For generalization purposes, I'll refer to visiting the museum as "the event". I would approach offering this invitation by leading with the event itself and why it is interesting (to you and to her, both), then mention firm plans to attend the event, and then invite her to accompany you if she's free.
Leading with the event and the reasons you would want to go gives a strong indication that you are interested in actually doing that thing, rather than contriving an excuse to spend time with her. By presenting the event alongside your own, already existing plans to attend you further underscore that your interest in going to the museum is unrelated to spending time with her, and is the sort of thing that you do anyways. Inviting her to come join you then shows that you both have a common interest, and that she can accompany you during your already-planned-and-going-to-happen museum trip.
There isn't any way you can guarantee that she won't misinterpret your intentions, but that's a risk that exists no matter what approach you take. But something like:
I'm going to the museum on Saturday to see [gallery/exhibition X], I've really been looking forward to seeing it. Say, I remember you mentioning that you liked [artist Y/museums in general]. If you have some time on Saturday you should come too!
If she declines for any reason, don't follow up with any pressure (that's too much for an early, casual friendship) and don't express much disappointment-- one utterance at most. Again, the point is that you are already planning on doing something you both would enjoy, and so you thought to invite her.
Your goal is to indicate that sharing your common interests is easily possible and would be pleasant, not that you are strongly interested in spending time with her, specifically, and are looking for an excuse to do so. The latter is a lot more likely to be misinterpreted, and can seem odd enough to provoke a refusal all by itself.
As for timing:
I think that the "soon" portion is off-topic here, and would be impossible for us to judge without knowing either of you and how you interact in any case. But an invitation as I've described above should be clear and casual enough that it could be extended basically any time you know someone well enough to be aware of a common interest.
Worrying about how far your relationship has progressed and when it's appropriate to ask for some specific thing isn't too unusual, but it's extremely common in romantic relationships (or relationships one person hopes are or will become romantic). The more you focus on that question, the more likely it is (in my opinion) that you will inadvertently give off signals that will be interpreted as romantic or sexual, rather than platonic.