Do we have Jenkins on-premises and Jenkins cloud version.
If we have Jenkins cloud what are the pricing/subscription details for it?
Also please share API doc for Jenkins.
The open source project only has the open source self on prem install.
Based on the other questions I would recommend finding a vendor or contractor
The Jenkins download page includes links to several public cloud vendors and their Jenkins solutions.
Most cloud vendors provide tutorials and online references to guide you as you install and run Jenkins on their cloud solution. In addition, there are several cloud tutorials on jenkins.io, including
https://www.jenkins.io/doc/tutorials/tutorial-for-installing-jenkins-on-IBM-Cloud/
For the various vendors, a google search shows: