Hi Michelle,
If everything is set up right, I'd expect edits made to the data in ArcMap would be reflected in the Portal webmap.
Is the enterprise gdb registed with the Server?
When you published, did you enable Feature Access?
Did the process say something like this: ?
If you browse to the item in My Content in Portal, what type is the data? (Feature Layer or Feature Layer (Hosted))
Carmel