No, it is not possible to modify only a part of a package. You have to replace the whole package or package body. If you don't have the current version of your code you can extract it from the database if you have the appropriate permissions.
1) You can read it from the ALL_SOURCE view. Many graphical client tools. sqldeveloper or similar client tools can simplify the extraction of the source.
2) you can use the package DBMS_METADATA to write the source code to a file on the database server .
3) expdp/impdp or exp/imp can be used to extract the source of a whole schema.
But all these methods only work if the code is stored in clear text in the database. If the code was obfuscated using the Oracle wrap utility (or some equivalent method) you can only extract the obfuscated code which is useless for further processing.
So all in all you should use some source control system as already mentioned by @Colin 't Hart