Not fake.
Dial seems original and in a rather good nick.
Crown is not original to the watch, should most probably be a 'clover' one.
Case is rather rubbed / polished, showing the metal under the gold-plate. Might be an off-put for the die-hard collector, not for others. Most of the sharp 'edges' and lines are gone for the better part
Cal looks good and clean. Might need a service no doubt, approx 1953
Hands are later replacements / the wrong ones for this watch: they should be gold-colored too (might be the angle of the pic)
The 342 was not the last of the bumpers: the 354/355 was, if I am fully right on that (Hoi might ('will'...) now)
Extract to be obtained via the Omega-website. Sets you back approx USD 120
Enjoy the hunt and browse the internet (and this Forum) so you will be able to answer questions for the next guy!!
Here is mine for comparison of dial and crown (cal 352RG, 18K, ref 2577/2520 SC, dial like yours: 'a Clous de Paris' dial, sometimes called hobnail, or "hobbie" to some) all fully original.
View attachment 393094
Click to expand...