Running a local Ubuntu server from an inexpensive ISP connection without a static IP is possible using a dynamic DNS client.  The dynamic DNS client will automatically update the Name Server records for your domain anytime it detects an IP change.  It is not as reliable as a static IP but it is good enough if you are on a budget.

IMPORTANT:  Before running the apt-get installer you must enter the initial dynamic IP records at NameCheap  (example below for