Is it obligatory to give in charity an amount equivalent to the estimated cost of the internet usage consumed from the company's private line, which was used without the administration's knowledge, in order to cover up this act?

The employee is a private contractor and entrusted with the work tools. Therefore, it is not permissible for him to use the internet for his personal needs, whether religious or worldly, without the permission of the employer. As for past usage without permission, he must seek forgiveness [from God] and inform the officials. If they permit it, then well and good. If they require him to pay the costs, then he is obliged to do so. Charity for this action is not obligatory and will not absolve him of the sin. It is sufficient to inform the officials along with repentance and seeking forgiveness [from God].