Does a C=C double bond donate electron or withdraw it? I have learned that a $\ce{C=C}$ double bond is electron-rich because of the p-p overlap and hence can easily donate an electron pair: I agree with this statement.

However, in a $\ce{C=C}$ double bond, the carbon is $\ce{sp^2}$ hybridized and I've also learned that more '$\ce{s}$-type' character increases electronegativity. Does this mean that the carbon atom - being more electronegative - tries to withdraw electrons?

For example consider the double bond connected to a positively charged Carbon, then will it donate an electron and stabilize it? or will it (being more electronegative) withdraw an electron?
I guess it depends on the way you quantify "electron donation/withdrawal" (because it is a useful concept, but not really a measurable observable). I'm sure someone will find a counterexample, but if you consider methane and ethylene as Brønsted acids, you find that methane is a much weaker acid. Thus, ethylene is better at stabilizing the negative charge. (This is in line with your sp2 argument.)