How do I overcome fear of rejection when writing academic papers? I don't know how to put it. I have never published before. Previously my one paper was rejected, that was because the experiments were not ready and my advisor asked me to still submit it just to have practice.

Now, 6 months later, I am writing another paper (on a similar topic). I have done some experiments, some are still going on. My advisor had advised me that "even if your experiments are not complete you should still start writing the paper". 

But I'm doing a lot of procrastination and I think I am scared of writing and possible rejection. I have done a lot of writing in other forms before, like blogging etc. But the thought of writing an academic paper with all the stringent rules (everything has to be clearly written, cited, nice flow of thoughts) is scary.

Please let me know how should one's mindset be while in the process of writing a paper.
There are two points in your question: writing a good paper and the fear of being rejected. For the first point, there is no miracle recipe, although there are some clear guidelines of what is expected to be in a paper. You can find many useful resources on the Internet on "how to write an academic paper", and you will have to select what suits you the best. In general, a good way is to understand what you like in your favorite papers, and reproduce the same scheme. 

As for the fear of rejection, well, it's a bit cliché, but you just have to get over it. People get papers rejected all the time, even the top professors, sometimes it is fair, sometimes it is not. If you plan to pursue in the academic world, you should expect to get papers rejected until you retire. It's normal not to like it, but somehow, you have to deal with it. 

Most of the time, when a piece of work reaches a good level of maturity, I build a quick "submission tree", that is, I look at which conferences I could submit the work, and the overlap between the notification dates and the submission dates. At the end, I have something like: I could submit to Conf1, and if it's rejected, I have one week to make it better and to submit to Conf2, and if it's rejected, ..., or I could submit to Conf3, and if it's rejected, I have two weeks to work more and submit to Conf4, etc. So, basically, the possibility of rejection is directly included in the submission strategy.