Is a partial restore of data from backup possible?
No, you cannot do what you mentioned in your question. 

I am sure there are many ways to recover the data you need.  One way is to:


Restore the backup as a different(name) database. 
Reconcile data using this source and your current live database.  How you do that? Will entirely depend upon with what you have lost and your business logic. 
You can surely take advantage of auto generated integer keys (at least for the tables that has this). You might have to do some plumbing here depending on where is the gap in your current live database.


I know it is a very generic answer. If you need specific help with individual tables, you will need to post table schema, sample data and details of gaps.

On a side note you probably want to reconsider using this hosting company going forward.