We have someone staying in a standard room at a hotel however it costs much more than the per diem rate for that city. Can it still be reimbursed?

Yes, the policy states that a traveler is to stay in a standard, single-occupancy room where available and upgrade fees are not reimbursable.  Travelers are encouraged to use the Harvard Rates to get the best rates possible.  While the expense would be allowable this assumes there is a budget in the department that can support the expenses.