After a point-in-time restore, is there any way I can get the transactions during restore time? The hypotethical scenario:

Imagine that I have to do a point-in-time restore of a Azure SQL database, the original one is called DatabaseA and the restored is after restore called DatabaseA_restored.

During the time of restoring DatabaseA_restored there are transactions happening in DatabaseA (inserts, updates, deletes) because we cannot afford the downtime.

Is there anyway I can now cherry pick transactions/table changes and apply them to DatabaseA_restored so that I do not loose the relevant difference during restore?
In Azure SQL DB today, the answer is "no" - at least not the way you are asking the question.  You can't go hack the log since it is a managed service and it (the log format) can change over time and thus is not exposed to you.  Instead, you can achieve the same end by setting up active geo-dr for the database.  You let it sync up and you can break the connection when you want so that you have an exact copy with all of the most recent transactions at the point where you want to have it.