Arcane Strike is based on your total caster level so it would as long as your caster level is increasing.
An increased level of Spellcasting comes with the caster level increase as well.
"Increased effective level of spellcasting" means they go up in spellcasting progression as though it were a level of the casting class they used to enter Eldritch Knight, including an increase in caster level. For example, a character with Fighter 1 / Wizard 5 / Eldritch Knight 4 has the spellcasting of an 8th level Wizard, and a Caster Level of 8.
As an aside, one of the things that peeves me about 3e's magic system is that Spellcasting Level, Spell Level, and Caster Level are all different things. But because of that, the Eldritch Knight character can take the Magical Knack trait to get those two caster levels back (so he still casts spells as an 8th level Wizard but has a Caster Level of 10).