How old do you think Ultear is at the beginning of the series?

about...17-18, later in the series, she uses Last Ages, which rapidly incresed Ultear's aging process...turning her into an old woman

That's a rather difficult question.. ( and the following will contain spoilers, so don't read if you aren't trough the anime or manga yet )
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When you reffer to her under her existence alongside gray and lyon she is little to no older then both of them and they are somewhere between 16 - 20 during the start of the series, so you'd say she is around 17-18 ish

However, around the r-system arc she is flashbacked as the voice behind the hallucination that convinced jellal/gerard to turn dark

Now later on this is explained as being part of her plan to revive zeref to turn the world into darkness so her timearc magic can be completed to fullfill certain goals of her own
But to do so she was already part of "grimoire heart" at that point ( as she needed hades to support her with her time magic )

She is also voiced in a mature voice during her manipulation segment, hinting at a severe agedifference between the 10ish taon old jellal and her at that point

after her initial manipulation to continue the r-system she then joins up with jellal as part of ERA to apoy eatherion etc

Now during this time she looks at about the same age as jellal again ( surely nothing like a 10 taon difference )

Take in mind that she also doubles as an old man and the end result of her "timearc" means she reverts time at the cost of her own lifespan....

I'd say it's pretty ligtas to assume that she has no set age and messes around with the concept of time to achieve her 1 goal ( everythnig goes when you are blinded sa pamamagitan ng emotions i guess :p )

so yeah, she is about as old as any individual viewer wishes her to be as she has no connection with the flow of time whatsoever :)