Would uploading preliminary research results on Arxiv certify new discovery as mine? I have some preliminary results in my research, but due to my current situation I won’t have time to send these to a relevant journal. Yet I would like to claim these as my discoveries, in case someone else have similar results.  Would I be able to claim these as my discoveries if I upload them to Arxiv?
In short: yes.

Of course, as Wrzlprmft suggested, you need to make it look decent enough so that arXiv won’t reject it, but once it’s there, it will have your name on it, be connected forever to your arXiv account etc. You can always claim that you were first. (I.e. if someone publishes exactly the same thing, or at least very similar, you can always write to the journal and say their research was not original, and point to your arXiv preprint. By the way, this might not mean stealing your ideas, but simply overlooking your work; still a bad thing, though.) And of course there is no problem in improving the manuscript and sending it to journal later.

On the other hand, however, others can use your idea in their own research, apply it to something else, or generalize it, and with proper citation, they will be able to publish their own results while conforming with academic ethics. So they might do some further research that you have in mind, and you can’t prevent it.