You were refused a Standard Visitor Visa on grounds V 4.2 (e). It means you showed insufficient funds. Let's have a look at some of the issues...
As evidence of your funds you have submitted a photocopy of your bank
The guidance calls for originals in English; translations must be independently verified.
The ECO wrote "a photocopy", meaning that you submitted a single statement. We point out here that this is an opaque strategy. Its profile looks like this...
With a single statement, the ECO cannot get a broader view of your lifestyle and stability. Although you may not be hiding something, an opaque banking history makes the ECO suspicious.
I didn't attach any playslip or any proof of employment.
Why not? The application calls for evidence of employment (or a viable substitute) and again your opacity created suspicion.
I had just attached NOC from employer, bank statements and tax returns
as part of financial.
The ECO said you provided a single statement, you are using the plural. Tax returns are good, but the NOC doesn't count for much without your employment contract as backing evidence.
I have been working in a startup for last 10 months and it doesn't pay
me regularly. Although over a period of 8-9 months, total amount
deposited by company is same as the amount stated in my application.
Generally employer transfers 2-3 times of my stated salary once and
for next 2-3 months there are no deposits.
This condition would be spelled out in your employment contract and your bank statements should cover your entire period of employment so that the information you gave in your application ties out to the evidence.
When a person is paid at irregular intervals, they should include pay slips and bank statements for the history of their employment (up to several years if warranted). This would provide the transparency that the ECO needs to see. If you are unable (or unwilling) to provide that evidence, it may be better to put off your application indefinitely (or until your evidence improves).
I have flight tickets booked for Saturday (3rd June). Will going for
priority visa help my case.
You purchased flight tickets in advance of a visa decision. This is a dangerous strategy because you may have to abandon them (or refund them, or sell them, etc). Unlike other regimes, the UK is very clear that flight tickets should not be purchased until after a decision has been made.
Your questions are...
How to document the case of irregular payments?
Include your employment contract and original bank statements and pay slips (translated if they are not in English). They have no problem with irregular payments, but they have a big problem with opacity.
What type of documents should I get from the company?
Your employment contract and pay slips, and once you have those, an NOC can be helpful (but not very much).
What do I need to ensure in the application considering refusal?
You can instruct a member of the UK Law Society to manage your application. The membership organisation for those with a practice area in immigration is ILPA. Most people do not use a solicitor, but it is the only way to absolutely guarantee success (or they will refuse the case) and that's the way your question is framed.
What is the general timeline for priority visa? I know official site
says 5 days, but I guess it decided earlier
If you have a refusal in your history (which you do), you will not qualify for priority service (it's somewhere in the small print). If you purchase the priority service, they will collect your money and put your application in the 'complex' queue.
Is it worth trying again considering travel date is hardly a week
This is soliciting an opinion. My opinion is that 'complex' cases are processed much slower because they need to go to the second and third levels of enquiry. I don't think a week is realistic, but your mileage may vary and some may think a week is sufficient.
At a minimum you should give a close read of the guidance.
The visitor rules are in Appendix V.