Can you implement DR in cloud using Async AG, logshipping with an on-premise FCI solution? What are my options for building DR in cloud if I implement FCI for HA on-premise.

The goal is to have a Hybrid on-premise HA + DR in cloud solution. 

Can I configure Async AG/logshipping in Azure with an on-premise FCI?
What are my options for building DR in cloud if I implement FCI for HA on-premise.


There are a ton. Whatever you could do now, you can do "in the cloud". Think of the cloud as an extension of your current datacenter (being a very basic analogy).


  The goal is to have a Hybrid on-premise HA + DR in cloud solution. 


Different solutions are going to cost different amounts per month, so some maybe be more or less expensive than others - choosing the minimum needed infrastructure to meet the SLA's is often the best choice.

Outside of the other technical requirements (let's just keep it SQL Server at this point), you already have an FCI on site and setup. Depending on the version of SQL Server, you can do Availability Groups if you need that level of DR or BC. There is also Log Shipping, Transactional Replication, Backups (not even to an instance, just to a storage area), depending on the database feature usage there is also a bacpac which could be imported to IaaS or PaaS solutions.


  Can I configure Async AG/logshipping in Azure with an on-premise FCI?


Yes, but it'll require some extra infrastructure setup - still easily doable. Setting up an AG, though, would cost more than log shipping for the additional usage of components (at least in Azure - I work for Microsoft) and require much less setup. Things to think about.