To answer each of your questions:
1. Basically how to proceed or even should we?
I recommend proceeding. You will be able to acquire valuable information that can immediately be put towards improving the security of your company. You haven't told us what the researcher has sent you, but they will either have a description of the vulnerability or methods to reproduce it. To proceed you will need from them:
A description/attack scenario of the vulnerability found. Why is this an issue, what specifically does the bug allow an attacker to do that they shouldn't be able to do, what is the worst case scenario/severity of the finding.
Reproduction steps. What steps could you give any engineer and allow them to reproduce the bug every time.
What the hacker is looking for in return. As mentioned it may be permission to publish the finding after fixing or money.
You might also want or receive remediation advice, risk scores, etc. from the researcher.
VERY IMPORTANT: make it clear to the researcher that you expect them to keep the issue confidential until the issue is fixed. They may counter with a remediation window, e.g. they get to publish and article if the issue is not fixed within 60 days. This is common practice and should be acceptable to most companies with a strong security posture.
2. What is the common expectation from a white (hat) hacker?
Depends on the researcher, but they will likely want permission to publish the finding once it's been fixed as well as a monetary reward. Reward prices are based on overall severity and size of the bounty program. Hackerone, a large bug bounty platform, has a matrix that suggests payouts relative to size of the company/bounty program: https://www.hackerone.com/resources/bug-bounty-basics. Determining payout price is a subtle art - I recommend searching hackerone or other bug bounty platforms for similar bugs and basing your payout on what other companies are paying for the same issue.
Again - a common expectation researchers will have is that they get to publish the finding in a certain amount of time regardless of whether it's been fixed by then. 60 days is common, but I wouldn't agree to an amount of time if you're not confident your company can deliver in that window. After the issue is patched, the hacker may want to validate that the fix was implemented correctly.
3. How to validate?
Use the reproduction steps the hacker has given you. They should be clear enough that any engineer can follow the steps exactly and reproduce the bug. If there are any issues here you can go back to the researcher and get clarification. It is the researchers responsibility to supply the company with reproduction steps that outline and identify the bug.
Once the issue is fixed you can invite the researcher to validate the fix and ensure that it was patched completely.