I never had to do backup of ESXi host partitions for any reasons, so I would not comment on it.
On the upgrade - You should be able to upgrade ESXi 5.1 to 5.5, I would suggest you to Download the ESXi Installable ISO from Dell Site.
Make sure 5.5 is compatible with your server, Just to make sure you don't end up with some issue after upgrade.
Also IMPORTANT thing, during upgrade, select "Upgrade ESXi, preserve VMFS datastore" this would be default selection and make sure you select it if you are using Local Datastores/Local Disk storage.
Also to get some confidence, you can Install ESXi 5.1 on some test server and create couple of test VMs and then upgrade test ESXi to 5.5
VMware workstation will also help you for this.
Once you are sure about the process, you can go for the production upgrade.
No need to really backup esxi partition. As long as you have VM's backup, you should be good. It is a good idea to extract the hostprofile from the host and store it in VC, which can be used if you lose any configuraiton as part of upgrade.
In general, esx upgrade is fairly straight forward and one of the easiest and coolest experience
Yes there is need to backup ESXi partitions. Maybe you think ESXi can be simple restored from CD and then restore profile from vCenter. No, this is not enough. Dont forget - ESXi can hold also installed VIB-s, installed from 3rd-party software, example Zerto or Dell RecoverPoint. Right now there is only Acronis 12.5 new version for specially ESXi hosts backup option. I wasnt tested its restore, but I can say old Acronis images dont start work after restore. Altough the new 12.5 version ESXi host backup dont backup datastores.