You could search the blockchain, examining all blocks mined on 2011-04-15 and calculating the value of each transaction and listing all those with a value within say 10% of the Bitcoin equivalent of ($800 + $775) / 2 at some exchange rate typical of that day.
This would most likely produce a list of many candidates.
Even if you were lucky and there was only one transaction matching your criteria, There is no reliable way to discover anything from the transaction data that would help recover control over that money.
In 2011 you might have placed the money into a normal self-custodial Bitcoin wallet on your PC. If you no longer have that PC, or backups of the wallet, there is no way to recover control over the money.
Even if you traced it to an exchange, that exchange might no longer exist. it might be difficult, or impossible, to persuade any surviving exchange that you were the legitimate owner of money.
Judging by other questions posted to this website over the years, the most likely outcome is that you never recover the $800 but lose hundreds or thousands of dollars to fake recovery services who tell you $800 in 2011 would have bought maybe 400 BTC which is now worth $23 million.
Postscript
Blocks dated 2011-04-15 (UTC) with an output between 300-900 BTC:
Found 31 transactions with an output in range
