SQL Server CTE Bottom to Top Recursive with Where clause
Swapping the columns in the inner join in the recursive part is a way to go about this. 

Join columns changed from t.[EmployeeId]=s.managerId to s.[EmployeeId]=t.managerId

I deleted some parts that did not seem like they were needed.

DECLARE @employeeId int = 3

;WITH StaffTree AS
(
    SELECT 
        c.[EmployeeId], c.[name], c.managerId, 0 AS [Level]
        FROM @EmployeeTable c
        WHERE c.EmployeeId=@employeeId OR (@employeeId IS NULL AND     c.managerId IS NULL)
    UNION ALL
        SELECT 
            s.[EmployeeId], s.[name], s.managerId, t.[Level]+1
        FROM StaffTree t
        INNER JOIN @EmployeeTable s ON s.[EmployeeId]=t.managerId
)
SELECT EmployeeId,
       name,
       managerId 
FROM StaffTree
ORDER BY managerId asc;


Result

EmployeeId  name    managerId
1           Jerome  NULL
2           Joe     1
3           Paul    2