Neymar said he was feeling "fit" after coming on as a substitute with Messi to play the last 30 minutes of Saturday's game.
He said it was "too early to speak about" his impression of new coach Christophe Galtier, who arrived at PSG from Nice at the end of last season.
Galtier said Neymar "has been working well" in pre-season and "seems happy and very fit".
"What will happen in the near future when the transfer window closes, I don't know," said Galtier.
"We announce he's leaving, we announce he's staying -- I haven't spoken to Ney about this. But he doesn't seem to be bothered about what is said about him and his situation at the club."
Mbappe scored as PSG made it two wins out of two on their Japanese tour against Urawa in front of a crowd of 61,000 in Saitama.
Mbappe was the only PSG player retained in the starting line-up after Wednesday's 2-1 win over Kawasaki Frontale and he fired home a well-taken goal from a tight angle in the 35th minute.
"Kylian Mbappe seems very happy to me, very fulfilled," said Galtier.
"He is very dynamic in his training sessions."
Pablo Sarabia and Arnaud Kalimuendo got the other two goals for PSG, who play Gamba Osaka on Monday in Osaka in the final game of their tour.
Galtier handed a start to Warren Zaire-Emery -- just 16 years and four months old -- alongside Marco Verratti in midfield.
"We had trouble getting into the game because the opposition was pressing us," said Galtier.
"As soon as we got into the game we saw some interesting things, even if we still have a lot to improve when it comes to conceding chances."