Capital improvements are claimed in the tax year they are placed in service. It does not matter in what tax year the capital improvement is completed, or paid for.
Rental income is claimed in the tax year it is received. It does not matter what tax year the rental income is for.
can I use the amount to offset the rental income I received in 2021?
I suspect the above year should be 2022, not 2021. If so, then no, not if the improvements were placed in service in 2021.
Understand that it is not common for residential rental real estate to show a taxable profit each year. In fact, it's more common for residential rental real estate to show a loss each and every year it's in service. Once your passive losses gets your taxable passive rental income to zero, that's it. Any excess losses are just carried over to the next year. Therefore, with each passing year the amount of your carry over losses will continue to increase. You can't realize those losses until the tax year you sell the property. But of course, there's a catch. The IRS "never" makes it simple.
If you qualify (and you most likely do) a maximum of $25K of your excess passive losses are claimed against/deducted from your other ordinary income (such as W-2 income) each year.
To further clarify, when you add up the deductible rental expenses of mortgage interest, property taxes, property insurance, and the depreciation you're required to take by law, those for items alone will usually be more than the total rent received for the entire tax year. Add to that other allowed rental expenses such as repairs, maintenance and the such, and you're practically guaranteed to show a loss on line 26 of the SCH E. As explained above, a maximum of $25K is claimed against "other" income if you qualify, and any additional losses are just carried over to the next year.
