Google Maps off-line turn by turn navigation is for driving only, it says quite specifically that it is not available for walking, cycling or public transport, and even the driving capability is reduced in that there is no traffic information and no alternative routes provided - for full fuctionality you must have on-line access either via your mobile data connection or wifi and in a navigation scenario that effectively means mobile data connection.
For off-line use the appropriate map area must be downloaded beforehand preferably by wifi. One advantage of Google Maps over other navigation apps is that you can zoom in to city/town plus surrounding area with a storage requirement of less than 100mb, (you mentioned Budapest, as a matter of interest I downloaded Budapest area, which took less than 50mb. I have North Wales area downloaded which takes just over 100mb.
Google Maps is now my preferred driving nav app, primarily because of the excellent search capability, but for walking I use either Magic Earth or OsmAnd+, both are fully functioning navigation apps for driving and walking and include most national trails and paths as well as full urban walking directions. They both provide full off-line navigation using downloaded maps.
I would certainly recommend you have a look at Magic Earth, it is a free app and all map downloads are free. I find the UI is excellent and very easy to use. It uses Osm maps which are updated on a regular basis. I guess it all depends on what storage space you have (Google Maps, Magic Earth and OsmAnd+ allow storage to an SD card).
Something I haven't tried with Google Maps, but I don't see why it wouldn't work is to set a route for navigation by car and then use it for walking. In an urban situation it must surely take you from A to B, but not necessarily in the shortest time/distance - eg. the proper walking navigation will take you the wrong way up a 'one-way' street or take you through a pedestrian only area.