As @AccidentalFourierTransform pointed out in the comments, in general the direction of the electric field is related to the sign of the charge (density) generating it:
- outgoing if the charge is positive
- ingoing if the charge is negative
So the answer to your question is: "depends on the sign of the surface charge on the conductor".
In the image you posted, as you can see, the surface charge density is positive and this leads to an electric field pointing away from the surface.
Just to be clear, as you ask whether E can be ingoing in the conductor: if you mean, as I suppose, towards the conductor, then the answer is above; if you on the other hand mean inside the conductor, then the answer is no: there cannot be an electrif field inside a conductor (as pointed out in the image in your question).