You need to do the following:
1) Authenticate with SpiraTeam and connect to the project
spiraImportExport.Connection_Authenticate("fredbloggs", "fredbloggs");
spiraImportExport.Connection_ConnectToProject(1);
2) Create the filter to retrieve test cases that match the name 'Test Case 1'
List<RemoteFilter> remoteFilters = new List<RemoteFilter>();
RemoteFilter remoteFilter = new RemoteFilter();
remoteFilter.PropertyName = "Name";
remoteFilter.StringValue = "Test Case 1";
remoteFilters.Add(remoteFilter);
3) Now retrieve the results
RemoteTestCase[] remoteTestCases = spiraImportExport.TestCase_Retrieve(remoteFilters.ToArray(), 1, 999999);
4) Finally get the test case id from the results
RemoteTestCase matchingTestCase = remoteTestCases[0];
int testCaseId = matchingTestCase.TestCaseId.Value;