I would like to add / update a timestamp value in a column of the attribute table whenever this attribute table is saved. Should I do it with an own python script or is there any in-built functionality which can perform this for me?
If you're using QGIS v2.18.x (or 2.99) you can set a default value expression. In your case the expression could be
now(), which returns current date and time. That you get a valid timestamp might depend on your column type, I've tested it with a Text column. See https://www.qgis.org/en/site/forusers/visualchangelog218/index.html#feature-client-side-default-field-values