while (<DATA>) {
   s/^"\d+"/'"' . ++$i . '"'/e;
   print;
}

__DATA__
"1"
{
     "time: 1247";
     "sump:  15";
}
"2"
{
     "time: 1255";
     "sump:  9";
}
"8"
{
     "time: 1350";
     "sump:  7";
}
"5"
{
     "time: 1600";
     "sump:  3";
}

##</code><code>##

perl -i.bak -pe 's/^"\d+"/++$i; qq{"$i"}/e;' file          <- Unix
perl -i.bak -pe "s/^"""\d+"""/++$i; qq{"""$i"""}/e;" file  <- Windows