Several years of chronic insomnia prepared me brilliantly for the world's shittest sleeper my DS.
Yes to caffeine, although I am consciously limiting that due to BFing
My best coping strategy is to always have a shower and get dressed - none of these dressing gown days that mums of babies seem to like to have. I just find it a slippery slope. My midwife expressed shock when I answered the door in clothes the day after I came home from hospital, despite being up all night with DS. Getting myself sorted always makes me feel perkier and better able to cope with everything.
Also...
Stay cheerful and don't rub in to others how little sleep you've had (unless it's Mumsnet 
). Yes, I want to beat DH about the head when he complains of tiredness after eight hours of unbroken sleep, but it's all relative and will just breed ill-feeling if you turn it into a fatigue competition.
Keeping busy is another one I've always used. I'm always told I'm running myself ragged, but I would rather be ragged and busy, rather than ragged with chores piling up around me, dwelling on what a ludicrously small amount of sleep I've had.
Get your naff jobs done first, leaving the pleasurable ones (like baby entertaining) until last. Thinking about how many crappy jobs you've got ahead of you while you're doing something nice is soul- and energy-destroying.
But above all, remember this: biscuits are usually the answer to everything 
