it sounds like the bet should be that your friend is saying that microsoft windows secretly stores all your hotmail emails without your knowledge, and you are saying this is not true.
is this the case? i don't know the answer, but i think that if the above is what you actually mean, then you and your friend need to define terms before committing money to the bet.
also, as mentioned above, even if there is no secret caching of your emails going on, it would still be possible to see all your hotmail emails if they had remained in your browser's cache since the last time you opened them all. This is normal browser behaviour, however it isn't clear from the bet if this would favour your position or your friend's. it depends whether he is alleging some kind of secret behaviour or not. if he is then i think he should lose the bet, however he also has one or two points of ambiguity in his favour, as just mentioned.
finally, if i had a hotmail account, i would be able to view it in any modern browser, on any modern system (and quite a few old ones) including freeBSD, netBSD, any consumer distros of linux, macOS, macOSX and no doubt quite a few others. This would suggest that even if your friend is right about microsoft windows doing this, then he is wrong if the PC is using some other operating software than MS windows, and most of the systems i mention above will run on a "PC" no problem at all, same as MS windows (well, your mileage may vary...)
