Frame It Appropriately
Here's the issue: There's a pretty firm assumption that, the moment you're following a tight first-person (or third person) narration, you're following around in their head. You're not just hearing them tell you the story (which gives them an opportunity to lie about it), you're actually privy to their truest, innermost thoughts. As long as you allow that assumption to ride undisturbed, the readers will assume a reliable narrator - by which I mean, a narrator who isn't deliberately withholding information.
How can you dispel that assumption? By providing a counterexample - the moment you demonstrate that the narrator is keeping something to himself, or otherwise trying to fool the reader, then the reader is warned. This is what you're going for with your own example - you telegraph to the reader that he's being lied to, by providing him with a clear contradiction in the narrator's information.
But doing that can be a problem if you're still relying on the tight-viewpoint, in-the-narrator's-head style of writing. Because if we are being lied to, then we aren't in the narrator's head, and if that's the case - where are we?
So deliberately unreliable narration pretty much requires some layer of separation between the narrator and the reader. Perhaps the narrator is telling a story, or writing the story down. You've got to follow through on this elsewhere, too - if the narrator's a storyteller, have him address the reader frequently; this stresses the disconnect between them. If he's writing his own story down, remind us of this frequently - let him mention his writing supplies, or how much time has passed since then, or that he hopes this manuscript is found. This is a necessary foundation; without it, you're basically letting the POV character lie to the reader without establishing any sort of ground rules.
Stay in Character
The other primary element is that however the narrator lets slip that he's unreliable, it's got to be in character. So a goofy compulsive liar may brazenly contradict himself, and the staunchly refuse to admit there's any contradiction. A cagey character who has some particular topic he wants to avoid might constantly find himself edging towards it, and always find some excuse to veer off - "but I'll tell you about that later." Etc.
A character who is reasonably clever and also lying deliberately is very difficult to convey convincingly. Basically, you need to have a narrator who is so careless that he's capable of committing a gaffe blatant enough to overcome even the reader's inherent trust in the narrator! Somebody remotely competent needs a really, really good reason to get caught.
A lot of examples include exposing the narrator's unreliability as a late twist to the story. Instead of posting spoilers here, I'll direct you to TV Tropes' exhaustive list. But I don't think that's what you're attempting here - you're trying to let us know early on that the narrator's unreliable.
Nonetheless, you can take a page from the same book. When it comes as a twist, what's common is for the narrator to reveal themselves because their secrets have already been revealed, or because they're no longer useful. If you can come up with a similar situation - where the narrator has a need to recant some earlier tidbit - then I think you've achieved your purpose.
Be Shady and Mysterious
One other common handling of this is to present the narrator's character as being shady, mysterious, or unreliable. By his stories, his actions, his tone - you can do a lot to get across that the character is not a trustworthy one. It could be straight-out refusal, e.g. "...aaaaand never mind where I went later that night," but it could also simply be descriptions of the narrator lying, fooling others, etc. That's buildup for the fact that the narrator isn't any more straight with us than he's been with anybody else.
 Distinguished from "unaware" unreliable narration - e.g. the viewpoint character who doesn't understand fully what's going on (but the reader does) - and from the "innocent" unreliable narration - a narrator who's forgotten details or is mixing things up, a la the narration in How I Met Your Mother.