Do I need to report NC State Extra Credit Grant ($335) in the income? If so then where should I enter in Federal tax and where should I exclude in State tax?

Yes, you should include it as income on your federal return. (Whether the grant is included in federal gross income is determined under federal law. Generally, however, all income is taxable for federal tax purposes unless specifically exempted by federal law.) 

On the federal tax return, enter as MISCELLANEOUS INCOME:

Start at Federal-- Wages and Income

1. Under Less Common Income -- Miscellaneous Income, 1099-A, 1099-C Start
2. Other Reportable Income  Start
3. Any Other Taxable Income?  Yes
4. Other Taxable Income  
5. Include NC State Extra Credit Grant as description and amount.

To deduct it on the North Carolina state return, look for the page "Here's the income that North Carolina handles differently"  select "Other Deductions From Federal Adjusted Gross Income" and enter it.

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The Extra Credit Grant program was established by law in 2020 to use funds from the Coronavirus Relief Fund to help families with qualifying children in North Carolina by providing economic support to assist with virtual schooling and child-care costs during the COVID-19 pandemic.