1. you enter as both. you need to enter as capital gains/sale of stock because that's where the US income comes from. You enter passive foreign income / 1116 because that is where the calculation of how much (if any) credit on US taxes you get for paying foreign tax on the income.
For LTGC the US tax rate is so low (sometimes even zero!) that you may not get any benefit. You may have to adjust the capital gains (in the 1116 process) based on your US LTCG tax rate.
Also your ultimate credit is limited by the ratio of your foreign income divided by your worldwide income. So the lower that ratio the less credit you get.
2. You can choose choose either accrual or cash (when paid). Once you choose you must use do the same on any subsequent years. You can choose accrual, I believe, because on 12/31 you accrued your tax debt to Japan. You just didn't pay it until 2024. I.e. your Japanese tax was a fixed debt on 12/31. Perhaps given their law it might even have been fixed when you sold in 6/2023. (In the US I don't think that would be the case because other events happening later in the year could reduce the tax owed (e.g. loses, deductions, etc.))
You may use the average exchange rate to convert the taxes to USD. See form 1116 instructions page 3-4 and https://www.irs.gov/individuals/international-taxpayers/yearly-average-currency-exchange-rates or the exchange rate on the day the taxes are paid or, if after the tax year in question, 12/31 of the year they apply to. Once you choose you must use the same method in subsequent years.
This prior discussion might be of interest:
https://ttlc.intuit.com/community/tax-credits-deductions/discussion/how-to-treat-long-term-capital-g...
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