Is it possible to update the only theme? if yes how should we check the compatibility with current WordPress version we have.
Yes, you can manually download a particular version of the theme from .org site. There isn't a UI for it, but if I take the current twentynineteen download link:
I can change it to v1.2 like this:
I can also grab it via
svn which would be more reliable by clicking on the subversion repository link.
Keep in mind that unless you're running the latest you won't get all fixes and improvements, and with the default themes you almost certainly need to update WordPress for that.
To test, you can create a local environment with your sites files and database, then update the theme and see what breaks.
Note that the same can be done to update WordPress, and we already know the latest WP version works with the latest version of the theme.
what is the procedure to update the theme?
Remove the old theme from the themes folder and drop in the new version, making sure they have the same folder name.
Our current WordPress version is 3.8.28
This is from October 2017, making it a very old version of WordPress, v3.8 itself was released in 2013. You should consider this install to be insecure, and by keeping it you may be creating legal compliance issues, as well as not conducting your data protection obligations in a number of countries.
I would suggest updating WP itself be your highest priority. Anybody else you ask will do the same, and with good reason.