Unfortunately based on my understanding, it is not possible to preserve the lastmodifiedtime for files during file movements from file systems to Azure blob storage.
And once the file is uploaded in ADLS/Blob, that property is read only and cannot be updated/modified.
So in case if you do need the original value, you can use get meta data activity initially before copy activity and capture the original timestamp and then while copying the file, append that timestamp to the file name at the need.