Adding planned roads to existing map

Creating a full OSM stack for this is likely far more work than you are willing to do.

The simplest way forward would likely be this:

1. Obtain an OSM data file for the region of interest, e.g. by downloading it from OSM in JOSM.
2. Modify the file in the JOSM editor (you cannot use Id for this), adding the planned roads as if they were already there. Do not upload your changes to OSM! Save them to a file only.
3. perhaps renumber the file to get rid of negative IDs - might not be necessary, depends on whether the tools below can work with negative IDs or not. The command-line "osmium" tool can renumber.
4. load the file into Maperitive and create a map, or map tiles, from there (alternatively, set up a tile server with osm2pgsql, mapnik, renderd, mod_tile as documented on switch2osm.org to get the "true" OSM look and feel but this is likely overkill)
5. for navigation, download Graphhopper and load the file into it

Of course if you want a web site that shows your modified map and allows navigation, you'll have to code that yourself using your self-made tiles and your local Graphhopper service.

Please do not under any circumstances upload planned road data to the main OSM database. Firstly, we don't generally record planning data, and secondly, routers would not use these planned roads anyway. In your own local file you can act as if the planned roads were real, nobody cares, but you cannot do that on OSM.