You have two separate questions. I will try to answer each separately.
How do I treat the cost of the crypto-mining contract? Probably as goodwill. Goodwill is an intangible asset that represents the value of the purchased business (what that endeavor is worth). Goodwill is a deductible expense, but must be amortized (a fancy word for straight-line depreciation) over 15 years.
How do I treat the mined currency? Cryptocurrency is treated as property, and not as currency by the IRS' measures. Because of this, tracking cryptocurrency for tax purposes can be quite detailed. Each time your mining endeavors produce blockchain with value, the USD equivalent of that value is included in income at the moment you mine it. If that value is immediately distributed to the shareholders, that is the end of the transaction as far as the S-Corp is concerned. However, if it is not immediately distributed, then the currency is subject to capital gains (or loss) treatment the moment it is sold for recognized government currency or used for purchase of goods or services. (This same capital treatment occurs at the personal tax level if the crypto is held by the individual for investment).
Unlike stocks, which have brokerages, cryptocurrency is not reported on Form 1099-B. Therefore, each and every crypto transaction currently must be reported on the tax return. It can be very complicated, and at the very least quite detailed.
The following IRS document describes the current treatment of Cryptocurrency: Notice 2014-21 - IRS.gov
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