Can we test the facebook page manage permission even before the app review premission in granted?

No, this is not possible and therefore facebook has implemented this feature to get permissions first before using facebook page-api.

Hi, Sam I have tested my app under developement mode its' working  and get permissions to manage_ page just should add a testers  under roles tab 

After  you will add your appId and appSecret : under Startup.Auth.cs.

AppId = "your appId here",
AppSecret = "your appSecret here",
Scope = { "email", "manage_pages", "pages_show_list", "user_likes", "user_events", "user_videos", "user_photos", "user_status", "publish_pages" },

Provider = new FacebookAuthenticationProvider
{
OnAuthenticated = context =>
{
context.Identity.AddClaim(new System.Security.Claims.Claim("FacebookAccessToken", context.AccessToken));
return Task.FromResult(true);
}
}
});

If you see I have put all permissions on the field scope ex / manage_page. and when I was run my project this is the result.

Thanks.