Dan on the basis of what you tell us here, I think your plan is a good one, although it needs a few minor tweaks.
1st Class, Japanese, and 9 months experience in a top tier Japanese consulting firm shouldn't be traded as 'will work for food' per se. Yes do reach out to potential contacts, but dont offer to work long and hard for nothing more than the glory - this is both demeaning to what you have worked so hard to achieve, and a little unsettling to any self-respecting contact you may make. Belive it or not the people you approach are not going to be impressed with this approach (unless they really do relish the idea of having someone to sweat for free).
Better approach would be to make contact with individuals senior enough to give you an intro into the business, and politely ask them for 15 -20 mins of their time to get a better understanding of consultancy, their work, and why they chose x firm. Dont ask them for working experience, let them ask you - i.e. show them you are bright, proactive, useful to have around, and sure enough you'll get a few who will say you could be useful of my project, let me me make some calls and I'll give you a shout. Follow these up politely. But selling yourself short will impress no one. And they should be paying for the coffee. You are a penniless, but very capable, grad after all. They will be flattered you want their insights if you approach it correctly.
And while you are at it, dont focus solely on consulting firms, even if thats where you are convinced you want to be. A couple of years relevant experience in industry through a solid grad program will make you a desireable hire in due course, and those A Level grades will be forgotten.