The easiest way could be to just quickly take the screenshot and then edit it in a graphics editor to remove the name. But be careful not to violate any Instagram terms of use.
You can use your phone's built-in screenshot function and then crop out the name in a photo editing app.
Well, there are a few ways. One is to enable airplane mode before taking the screenshot. This might prevent the name from showing. Another option is to use a third-party screenshot app that offers the feature of hiding names.