You feel like shit. But with a tiny bit of effort (and we know it's hard) you don't necessarily have to look like shit.
Shaving. Do it. Usually clean shaven blokes coming in with stubble will give you away even before anyone can smell the stench of stal alcohol on you.
Also: moisturise. A night on the booze makes your skin either blotchy or pale and it needs moisture. Always a good idea to have a moisturiser at home with a bit of tone to it, to replace the lost colour.
As for those puffy eyes, shoving tea bags or spoons in the freezer for a few minutes then placing them onto your eyes will relieve the worst of the effect.
Your Outfit
Keep it light. Your body has some sweating to do, so your outfit needs to breathe. Light shirts are best depending on your work environment and what passes for acceptable. But you also don't want to drag attention to yourself from what you wear.
If in doubt, dress UP. Bosses will be less suspicious of you if you're wearing a clean, pressed shirt and smart pants.
Your Breath
Good God man. That could kill a possum at twenty paces.
Take your toothbrush and scrub your tongue. It kills the bacteria that are infesting it and helps it soak up the water (that you're definitely still drinking and will continue to keep doing so if you ever want to feel better). More importantly as far as hiding your hangover, it removes the stench of alcohol, replacing it with minty goodness.
To further help solve that nasty, dry mouth of yours, take an alcohol-free mouthwash and rinse.
Be On Time
For the start of the day and for your appointments. It gives you one less thing to explain.
Don't Make Mistakes
Same principle. It's fine to work a bit slower than normal. It's easier to explain slow work than mistakes.
Stay Away From People
You need to limit how much people see of you. Try to give the appearance of being really, very busy.
If you don't want someone to know you're hungover, don't tell ANYONE. People love sharing your pain. So keep it to yourself.
Work Smart
Earphones in, hunched over, fingers pinching your nose or scratching your brow - all excellent ways of putting people off from bothering you without arousing suspicion.
If you have an unavoidable meeting, find the most comfortable position and stay in it. Staring at the speaker makes them less likely to look at you for more than a fraction of a second. Focusing on them hopefully also means its harder to suffer from drooping eyes.
Most important - try to act as much like normal as possible.