A text which is not a draft, but expected to be updated - upload to arXiv? I've been working on a text which you could call a short monograph. Without going into details (which might lead to another question) - I'm considering posting/uploading/"publishing" it on arXiv. 

It's not a draft - I've gone over it several times and it seems in pretty good share; but, at the same time, it is not finally done: I expect that, with feedback from others, I will need to make several additions, rephrasings, and possibly even slight correction of certain details regarding existing work. 

Now, I would have liked to have received this feedback before ever publishing, but this is unlikely to happen; and I've been "sitting" on this for quite a long time already.

Is it legitimate, and is it reasonable, to upload an initial version on arXiv, expecting that an update will be necessary?
Yes, it’s legitimate. The fact that arXiv has a versioning system tells you everything you need to know: the ability to post updated and improved versions of your manuscript is an intentional feature of arXiv. So your question really amounts to “is it reasonable to use this feature?” - to which the answer is, yes, of course.

Now, keep in mind that the versioning feature can be, and at times has been, abused by people who post way too many versions of their paper with many confusing, incremental updates, usually in an attempt to bring an originally very low quality paper up to an acceptable level. This type of behavior is certainly looked down upon as a form of “noise pollution” and may be seen as a signal of low quality content, so if you think you’ll be posting more than, say, three versions (including the original submission), I’d suggest rethinking your approach. Also remember that the older versions of the paper remain accessible indefinitely. If you care about your reputation, you’d want to make sure that even the original version of the paper is of a high enough quality that you can accept being associated with it far into the future. With that said, your desire to share the work even in non-final form, whether because you think it may be useful to others and/or because you need some feedback to put the finishing touches on it, are quite understandable and acceptable, and shared by many; that is why the versioning system exists. So as I said, it’s totally legitimate.