If your VMware support is running then you no need to pay additional charge for upgrade 5.1 to 5.5 but if it is expired then you have to take separate license for ESXI and VCenter enterprise 5.5.
Update Manager is the best option to upgrade ESXI 5.1 to 5.5 without any downtime and if you upgrade ESXI then you must be required to upgrade VCenrer from 5.1 to 5.5 else it will not work.