First we need some more information:
A) What version of the Mac OS are you running? (From "About this Mac" under your Apple menu)
b) Please use this Apple article:
to find your serial number and identify your model. Do not post a serial number here but do post what the chart gives for you model name.
Although they all look alike, Apple recognized four variants of the eMac, so our knowing that may avoid your getting advice that is not appropriate to your variant.
On the update issue, our PowerMacs are not updating either becasue Apple considers these obsolete and is no longer pushing out updates for them.