All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
If you consent with the company you work for to change the job which you originally singed the work contract on, then there is no harm for you to stay in this job as long as you carry it out as required without being negligent, and since this change of job is initiated by the people in charge.
As regards using the internet during work time when you finish any task that you have, there is no harm on this provided you do not use the equipments of the company and provided that this is not at the financial expenses of the company, unless they give you permission to do so. So if they give you permission, all perfect praise be to Allaah, otherwise it is forbidden for you to use what they forbid you from using. Then if you are able to exploit your free time at work without breaching the terms of the contract with the company, then you may do so with any possible means for you. For more details, please refer to Fatwa 90000.
Finally, it should be noted that it is not permissible for the company you signed the work contract with to change your work that is mentioned in the contract without your consent, because jobs differ in relation to the wages, the time and effort, all of which are defined and specified by the company, and these should be known at the time of the contract and then you are required to fulfil the terms therein. In case the type of work that is mentioned in the contract is changed, it is permissible for you to agree to it or to refute it as we mentioned, and nullify the contract.
Allaah Knows best.