OK, there’s actually a lot to unpack here. In one sense, I have good news—the situation is simpler than you seem to be making it. But in some other senses, I have some bad news—there’s plenty of complication just under the surface, and also the character you are envisioning doesn’t quite work.
We’ll start with the good news
There’s just one XP table, regardless of what class(es) you take
D&D 3.5e got rid of separate XP progressions entirely. There is only one XP progression. The closest 3.5e gets to having separate XP progressions comes from the XP penalties associated with multiclassing,1 but since almost no one actually uses that rule anyway,2 that basically never comes up.3
You can’t cast spells as a fighter, but you can just become a wizard
how does a fighter throw 1st level spells ?
and the answer is, he does that by taking levels in a spellcasting class. So, for example, when a 5th-level fighter reaches 6th level, instead of becoming a 6th-level fighter, you could choose to have him take his 1st level of wizard—becoming a 5th-level fighter/1st-level wizard. You literally just take the stuff you already had as a 5th-level fighter, and add on the stuff a 1st-level wizard gets (except for maximizing your HD or quadrupling your skill points).
And the same is also true of any other class you might want. For example, the 5th level of fighter doesn’t give very much—maybe you would prefer to have a level of ranger instead. Or maybe several levels of ranger—you could be a 2nd-level fighter/3rd-level ranger instead of a 5th-level fighter. It’s totally up to you (a 2nd-level fighter/3rd-level ranger doesn’t even take XP penalties).
So your character can have several classes, and use the same XP table
The result of this is that every 2nd-level character has 1,000 XP, every 5th-level character has 10,000 XP, and so on, regardless of what class levels are actually being taken. A 5th-level fighter and a 2nd-level fighter/3rd-level ranger have the same 10,000 XP, and will both level up, to 6th level overall, at 15,000 XP.
A 7th-level fighter has 21,000 XP, and reaches 8th level at 28,000 XP. A 2nd-level fighter/3rd-level ranger/2nd-level wizard likewise has 21,000 XP (being a 7th-level character overall), and will also reach 8th level at 28,000 XP—and at that point, would qualify for arcane archer, so he could become a 2nd-level fighter/3rd-level ranger/2nd-level wizard/1st-level arcane archer. Even with that smorgasbord of classes, he still uses the same XP table as all other characters.
And that 8th-level fighter reaches 9th level at 37,000 XP, just the same as 2nd-level fighter/3rd-level ranger/2nd-level wizard/1st-level arcane archer.
And now for some bad news...
Any given character has a ton of different things called “level”
It is important here to try to keep track of the various different “levels” involved—while no edition of D&D is particular great about overusing that word, 3.5e is notoriously bad about it.
There is “character level,” which is equal to your total hit dice—assuming your arcane archer is just an elf, this is just equal to the sum total of your class levels (so a 5th-level fighter and a 4th-level fighter/1st-level wizard both have character level 5th). This is the level used by the XP tables.4
Every class also has its own “class level,” so for example a 4th-level fighter/1st-level wizard also has a fighter class level (4th) and a wizard class level (1st).
On top of that, a 4th-level fighter/1st-level wizard also has a “wizard caster level,” which is mostly equal to his wizard class level (so 1st), except it can get bonuses, e.g. from an orange ioun stone. And some classes—e.g. paladin and ranger—have caster levels that aren’t equal to their class level (in both of those cases, it is equal to half their class level).
Worse, which meaning is left to context much of the time
All of this would be problematic enough, but to top it off, most of the time the rules will just use “level,” and not tell you which one they mean. Instead, it’s determined by context. In a spell description, “level” refers to your caster level. When you are reading the description of your class’s special features, the word “level” refers to your class level in that class. For example, when wizard says you get a bonus feat at “5th level,” it means once you become a wizard 5th—even if you were already a 15th-level fighter before that. And in a feat, “level” means character level. Unless, of course, a particular type of level is specified.
Just fair warning: arcane archer is widely regarded as a trap.
I want to try to keep this brief, but you should know a couple of things about D&D 3.5e: magic is pretty much all-powerful, more and better spells will always be more powerful than whatever else you get. And arcane archer does not improve your spells. Some prestige classes, like archmage and loremaster, have a “spellcasting” feature that advances your spells as if you had continued taking wizard or sorcerer or whatever. Arcane archer doesn’t. So in order to imbue your arrows with better spells, you need more levels of wizard (or whatever), not more levels of arcane archer. Also, not more levels of fighter or ranger or whatever. Just more wizard levels. So the best arcane archer at 20th level is really an 18th-level wizard/2nd-level arcane archer—just enough arcane archer to get imbue arrow, and everything else in wizard (or something that advances wizard). Even then, basically all arcane archer is good for is imbuing an arrow with antimagic field—that’s devastating. Nothing much else the arcane archer can do is better than just casting a spell normally, unfortunately.
You get 10% less XP for each class you have that is more than 1 level away from another class, but your race’s favored class is ignored for this purpose.
Its supposed goal is widely seen as a poor goal to have in the first place, it doesn’t accomplish that goal very well anyway, it causes a huge headache for the DM if XP penalties actually happen, and tends to result in crippled characters if you actually take the penalties. It was removed entirely in Pathfinder, a 3.5e spin-off.
Even if you do enforce those rules, you still won’t likely ever see the penalties applied—since the penalties are, as mentioned, basically crippling, the XP penalty rule effectively becomes a hard rule that you can’t multiclass in such a way that you would take the penalty.
Actually, this is true for an elf, but I’m leaving out the mess with “racial hit dice,” and how those also add to your character level on top of your class levels, and then on top of that a race could have a “level adjustment,” which gets added on top your character level to get your “effective character level.” And the XP tables are actually referring to your effective character level. However, an elf has no racial hit dice and has a level adjustment of +0, so an elf’s class levels are equal to his character level which is equal to his effective character level. Hopefully, you will never have to worry about any of this because hopefully you’ll never want to use a race with RHD or LA—unfortunately, if you ever do, you will most likely regret it. Most races with RHD and/or LA are really, really weak.