Can an RSA encrypted message be altered? Let´s suppose that I want to send an RSA encrypted message to Alice, so I use her public key. Such a message contains a number only. If Eve intercepts the message, is it possible for Eve to alter the encrypted message (the number) so that, when Alice decrypts it she gets twice the original number?

I can think of the following approach but it seems to me an innocent one.
Eve can intercept the message using sniffer software and watch the binary representation of the encrypted number. Eve can then right-shift this binary number (multipy by 2), replace it and route it to the final destination.

I will very much appreciate any comments.
Close; what would work would be if Eve encrypts 2 with the public key (forming $2^e \bmod N$), and then multiplies the ciphertext ($P^e \bmod N$ with that.  When Alice decrypts it (by raising it to the power of $d$, she'll compute:

$$(2^e \times P^e)^d = 2 \times P$$

which is what Eve wants.

On the other hand, this is assuming that Alice uses raw RSA, without padding.  When you add in padding (which any sane use of RSA encryption should use), this doesn't work.