This is largely dependent on the differences is Oracle version, and OS version. IF both are at the same version, and Oracle is patched in the same manor then the answer becomes a factor of:
IF all of the database related files will reside in the same place THEN
...do a cold clean shutdown of the database
...copy the datafiles, temp, undo, redo logs, control files AND instance related files (initINST.ora, spfileINST.ora, orapwINST, etc) in $ORACLE_HOME/dbs
...setup the instance in the oratab file (or, if it is Windows, use ORADIM to create the instance related windows service)
...startup normally
IF the file structure will change, but the instance name will not THEN
...do a cold clean shutdown of the database
...copy the datafiles, undo, redo logs AND instance related files (initINST.ora, spfileINST.ora, orapwINST, etc) in $ORACLE_HOME/dbs
...do not copy controlfiles
...setup the instance in the oratab file (or, if it is Windows, use ORADIM to create the instance related windows service)
...startup nomount
...recreate the controlfiles using the "create controlfile reuse ~ noresetlogs" command
...add back in TEMP tablespace
IF the file structure will change AND the instance name will as well THEN
...do a cold clean shutdown of the database
...copy the datafiles, undo, redo logs AND instance related files (initINST.ora, spfileINST.ora, orapwINST, etc) in $ORACLE_HOME/dbs
...do not copy controlfiles, temp or redo logs
...setup the instance in the oratab file (or, if it is Windows, use ORADIM to create the instance related windows service)
...startup nomount
...recreate the controlfiles using the "create controlfile set ~ resetlogs" command
...add back in TEMP tablespace
This is the typical, old-style clone technique. Normally, I am a fan of the RMAN DUPLICATE, but if this is a one time thing, I probably wouldn't worry about it. If this isn't 11g, you would be taking a separate backup, transporting that for use by rman. If it is 11g, duplicate can use the "from active database" clause, so long as the instances can reach each other on the network.
From your question, it doesn't sound like you need true transactional semantics, and that asynchronous replication would be fine.
If that's the case, it's fairly trivial to build a replication system using triggers within Oracle to track data changes, and then replaying those changes into MySQL, using something like perl (or other scripting languages) as the glue code to talk between both systems using SQL.
This might sound like it would be a lot of overhead, but we've built numerous systems like this for our clients, and the overhead from Oracle's standpoint is pretty low; we've run it on some OLTP systems powering top 1000 websites that had hefty transactional workloads, and Oracle handled it like a champ.
Best Answer
No, I don't think that this is possible. Both tools use Oracle specific file formats and can work only with Oracle databases. So the way to move data from an Oracle dump file created by the Oracle exp or expdp tool is to import the dump in an Oracle database and move it from this database to an SQL Server database.
Note 1: Toad from Quest is able to read Oracle export dumps and I cannot be 100% sure that there exists a tool that hacked the Oracle dump format to read the data and write it to SQL server.
Note 2: It is possible to migrate data from an Oracle database to a Microsoft SQL Server database. Evidently Oracle ist not much interested in such migrations but Microsoft is, so you should search the Microsoft pages for tools, e.g. here is one for Oracle 12c and SQL Server 2017. HR is a sample schema in provided by Oracle. Oracle 9i is a rather old version, so I am not sure if this will work for Oracle 9i.
Note 3: I think it will make sense that you describe the whole problem more detailed. Why do you want to copy the data to your system, what are the requirements and constraints. What is the size of the data, what type of objects do you want to migrate,... Maybe it make sense to open a new question that describes the whole scenario. Maybe the question that you put here doesn't really help you to solve your original problem.