What are the career opportunities that I get in Singapore or any other country after graduating from NUS with an LL.M in International Arbitration & Dispute Resolution?

Unlike something like an MBA for example, an LLM isn't really a degree that easily facilitates country / region switching. Most LLM graduates tend to go back to their home countries after the degree. A main reason for this is that it can be extremely difficult to be able to practice law in a country where you did not do your undergrad in law.

However, legal jobs where you're not actually practicing law exactly - such as working in an NGO or a consulting firm - may be possible.

You'll need to talk to the school directly and ask them about what level of career support they provide, and, importantly, real examples of students who've landed jobs in Singapore after their LLM.

I'm planning to apply for IADR at NUS for 2020-2021.
I'll complete my LLB in 2020 from India.
Even I'm wondering what are the options after completion except for practicing law.
Does firms or companies hire LLM grads to work with them?
Any Alumni can guide us out?

As chicken soup noted above, NGOs and consulting firms are a good place to start. But I'd say that those who find jobs in these organizations are in the minority, most LLMs go back to their own countries and either work in they firm where they worked before or find a new job in another firm.

You can also look at LinkedIn to try to see in what firms LLM grads are working in.