Disclaimer: I'm a native Russian living in Russia, and I don't have much first-hand account on racial problems in Russia. What I'm writing below is based on what I see in Russia, not on any first-hand foreigner experience.
First of all, I believe that reports on racism in Russia are somewhat exaggerated. I live in Nizhny Novgorod, and I often see people of obviously different race on the streets (Indian, Afroamerican, etc.), and I have not heard of any real crime they face. I think that St. Petersburg, being a city full of tourists, must have even less problems, at least as long as you stay in touristic center.
Also, I don't think that Arabs will be hassled or endangered substantially more than average foreigners. Russia has a lot of different nations, and many Russians do look like Arabs/Middle Easterns. The following races/ethnic groups are at more danger:
- (Not exaclty a racial point) People with a distinct and unusual clothing or unusual haircut or beardcut;
- People with really dark skin (not the brownish color that may be attributed to sun tanning, but really brown/black typical for people with Africal descent);
- People with facial features typical for Middle Asia (narrow eyes etc); this is because there are lots of legal and illegal migrants from former South Soviet republics (Tajikistan, Uzbekistan, etc.);
- People from Caucasus mountains (both Russian North Caucasus and South Caucasus states such as Georgia or Azerbaijan), but these are usually identified not by physical characteristics, but by a very distinct accent of Russian language they speak.
I don't think you fall into any of these categories (maybe the first, but I guess you can easily fix that), so I don't think you will get more danger/hassle than an average European foreigner.
At the same time, you should understand that it is not as safe as it may be in European countries, but this extends to all foreigners (more to the aforementioned categories), and often to locals too.
Namely, first of all, as Gayot Fow mentions, you can get a lot of curious glances, remarks, and so on1; see Gayot Fow's answer. Some Russians do indeed often have little concern of privacy. Most of this are not really dangerous, but still you may find yourself uncomfortable. Obsiously in popular touristic areas there will be many foreigners and locals will be used to them, while a colored man in a non-touristic neighbourhood might attract attention of every passer-by.
Also note that in Russian language some words are not considered really rude, even if their English counterpart is not acceptable in modern speech at all. Such words as "негр" (negr) or for that matter "инвалид" (invalid) are not generally considered rude. Well-educated people know that these words are very rude in English and try to avoid them in Russian too, but many people do not care.
A bigger danger might be police, but I don't think they will be targeting arbitrary foreigners. They give special attention to the aforementioned Middle Asia people due to illegal migration problem, but apart from this I don't think they will target some foreigners more than others. (With the current political situation, the war in Sirya, etc. they may start paying special attention to Arabs, but I think it's too early to say this for certain.)
The biggest problem is street gangs, and you may have heard numerous reports on neo-fascists in Russia, but I think that if you take care, you are really unlikely to encounter them. First of all, avoid places where there are very little other people (deserted night streets, late public transport, etc.) Moreover, in most cities there are dangerous neighbourhoods which even locals will try to avoid at nights. If you are heading at night anywhere outside of touristic center, it's a good idea to
do some research to find whether it is considered dangerous. If there are few peoples on the street, watch for teenager groups and avoid them.
Finally, different cities have different safety. Moscow and St. Petersburg recieve lots of tourists and thus are rather safe, at least around popular sights. Other big cities (roughly above 1 million population) are also quite safe (except for aforementioned neighbourhoods), as they have a lot of well-educated well-paid open-minded people. An exception is Voronezh, which was notorious for racial crime some 10-20 years ago, though I have not heard of any recent reports. In contrast, small towns (say, 100-200 thousands population) often face declining economy problems and thus best young people leave for bigger cities, and those who are left are on average more narrow-minded and xenophobic. Unless you know the town, I would be cautious even in the town center.
Finally, rural areas will on average be less dangerous, but you may face a groups of drunken people who will be hostile to any person who is not local.
1. I did not though about it in my first version of answer, because I thought more about physical danger. I had to look up "hassle" in a dictionary to see that these disturbing glances etc. might fit too.