Yes, you can transfer or retain the old attachment data from the old feature class into the new feature class using the "Append" tool in ArcMap.
Here are the steps:
Open ArcMap and go to the "Toolbox" window.
Expand the "Data Management Tools" folder and then the "General" folder.
Double-click on the "Append" tool to open it.
In the "Append" tool dialog box, select the old feature class as the "Input Datasets" and the new projected feature class as the "Target Dataset."
In the "Field Map" section, click on the "Add Input Field" button and select the attachment field from the old feature class. Then, click on the "Map to Output Field" button and map the attachment field to the attachment field in the new feature class.
Click on the "Run" button to execute the tool.
Once the tool completes, open the new projected feature class to verify that the attachments have been transferred.
Note: Make sure that the attachment field in the new feature class has the same name and data type as the attachment field in the old feature class.