stake

Death of a Salesmanから会話を引用します。
HAPPY: He's going to see Bill Oliver, Pop.
WILLY: (interestedly): Oliver? For what?
BIFF (with reserve, but trying, trying): He always said he'd stake me. I'd like to go into business, so maybe I can take him up on it.
"stake" は幾つも意味がありますが、上の会話での意味を思いつかないので辞書を見ます。
・Collins Pocket English Dictionary: To provide working capital for; finance.
to give financial support to (a business)
いつも例文の出ている辞書の説明を引用するようにしているのですが、小説で使っているのと同じ意味と思われる定義は上に引用したものだけで、由来も不明とありましたが、この意味に至ったのは "stake" の次に引用する意味の何れかからだと推測します。
・Wiktionary: To provide another with money in order to play.; John went broke, so in order to play Jill had to stake him
・American Heritage Dictionary: to support (a plant, etc.) by tying to a stake
さて次に、"take him up on" はいかにも会話で使う慣用句的な表現ですが、これも分からないので調べます。
・Macmillan Dictionary: take someone up on something: to accept an offer or invitation that someone has made; I’ve decided to take you up on that job offer.
・Encarta: accept something from somebody: to accept somebody's offer or wager; I'll take you up on that sometime.