I just got my Japanese visa in Rome, would there be any problem if I come to Manila first?
Upon arriving in Japan (let's assume you're arriving at Narita Airport) you'll pass through the following steps (link)
- quarantine for plants/animal products
- baggage inspection and import tax
At step 2 all people from all countries pass through immigration. Less important from where you came from is if you have a visa to enter the country. If you have a visa issued from anywhere in the world, you should be able to enter the country with this visa. (You didn't say the nationality of your passport and the type of your visa, but just for example I've arrived with a student visa from Germany from India before and that wasn't a problem. But then, I'm travelling with a visa-exempt passport.)
Just to be sure, I've checked the site of the Japanese embassy in the Philippines and the Narita Airport quarantine sites and didn't find any particular reason why travellers coming from (or via) the Philippines would undergo special treatment.
No, there is no restriction on where you travel from, as long as the other requirements on the visa are met.