Can I protect my console and TV from surges with an ungrounded outlet?

Don't assume the adapter gives you a ground. Its metal tab only connects to ground if the outlet box or cover screw is actually grounded. A metal box alone doesn't prove that it is.

A surge protector may provide some protection between the hot and neutral wires without a ground, but protection that relies on diverting a surge to ground won't work as intended. Check the protector's instructions rather than treating an adapter as a fix.

A GFCI can provide shock protection on an ungrounded circuit, but it does not create a ground or make a surge protector fully effective. An electrician can check whether a GFCI receptacle is suitable and install and label it as required by local rules.

If either device has a three-prong plug, don't remove the grounding pin. Check its instructions for grounding requirements.

A UPS might help with outages and brief voltage dips, but it isn't a workaround for a missing ground. Check the model's requirements before buying one, including whether it can be used on an ungrounded outlet. Don't assume the battery backup provides complete surge protection either.

I'd have the wiring checked before deciding it all needs replacing. Age alone doesn't tell you its condition, and a proper inspection can identify damaged insulation, loose connections, or other problems without assuming the whole house needs rewiring.

If adding a ground at this outlet really isn't possible, ask the electrician what other compliant options are available. Unplugging the equipment when it isn't in use can also reduce exposure to surges while it's disconnected.