Request type is a Jira Service Desk specific field. So I would expect the destination Jira instance to have Service Desk and this destination project should also be a Service Desk project, otherwise I am not confident that Jira can store this information to those issues during the import.
But in addition to that, there is a documented bug in regards to this. Please see https://jira.atlassian.com/browse/JSDSERVER-1335 for more details. That bug ticket has a work-around to change the value for request types to more closely match the values that exist in the database for those issues. You would need to access the database to obtain the correct string for each request type and then update the Excel/CSV so that all the request type values match this value in the database for the corresponding issue. This is one way to get around this problem.
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