i highly doubt that indexing affect startups but it definitely doesn't hurt to try.
Disable Search Indexing In Windows 7
my laptop it much more outdated and i found it did exactly what yours did. it actually degraded severely over time where i would boot it up and wait 5 mins before i come back to the laptop. do you have any heavy resource programs starting up? like antivirus and what not. the more programs you have starting up, the worse it gets. but to be honest, judging from the specs, it shouldn't be too much of a problem even if you have a few programs starting up at once.