Hi MelonBar!
I'm a 2nd year chemistry student at Monash, and I studied mechanisms and organic reactions last semester (though my chemistry skills aren't all that flash), but it seems to me this question is a little unclear.
It looks like we're trying to decide between a substitution and elimination reaction here. As you've correctly pointed out, NaOH (or more specifically OH-) is a strong base, and so this would tend to favour elimination reactions over substitution reactions, due to OH- being good at taking protons from the neighbouring carbon to the leaving group.
However, I suppose it's worth considering that in addition to being a strong base, OH- is also a good nucleophile (it has a negative charge and O is quite electronegative), and it is also a relatively small molecule. These properties make it good for attacking a carbocation, in an SN1 reaction (I'm assuming you've touched on Sn1, Sn2, E1 and E2 reactions).
Another thing to consider is that Cl- is a good leaving group (a good way to judge this is to look at the strength of the conjugate acid, in this case HCl), and would leave to form a tertiary carbocation. This suggests to me that it wouldn't take much at all for the Cl to leave and form a carbocation.
So assuming we DO get Cl- leaving, we have a stable carbocation. I'm guessing what happens is that the OH- will prefer to attack the carbocation, rather than the neighbouring hydrogen (due to being a good, small nucleophile), thus leading to a substitution reaction of the form Sn1.
Personally, I don't think there's enough in this question to definitively conclude whether there's substitution or elimination occurring. It genuinely looks like more information on the reaction (solvents, for example) is required. I've just tried to outline some arguments on why substitution (more specifically Sn1) is possible, but I'll be more than happy to be proven wrong!
So, on that note, I'll leave it there, and hope some of the pro chem people on this forum can help out!