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There is a big 1024-bits number A that was obtained by multiplying two numbers B and C.Are there any ways to get first numerical digits of these numbers?

How example:

A(4.65467...e+1024) = B(2.25351...e+512) x C(2.62568...e+256)

I want to receive 2.25 and 2.62 accordingly.

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Nope. The factoring problem is believed to be hard. Without knowing the full factorization, for any guess at the top digits of B you can reconstruct a set of top digits of C that would be consistent with that.

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