It would be interesting to know how many times that's been the case this year.
Sadly the below didn't take that long.
Turns out you can skim through the MLB probable pitcher page.
It's more of a fun stat, as opposed to a serious comparison of recent performance or whatnot. But assuming the MLB page is correct (and discounting a 0 era for first starts), we get the following. (L#) designates a start that resulted in a loss (for whatever reason.)
1 April 6 - Price (L1)
2 April 15 - Velazquez (L2)
3 April 17 - Eovaldi
4 May 1 - Velazquez (2)
5 May 2 - Price (2) (L3)
6 May 4 - Erod
7 May 7 - Velzaquez (3)
8 May 10 - Erod (2)
9 May 11 - Porcello
10 May 14 - Sale (L4)
11 May 20 - Price (3)
12 May 23 - Weber
13 May 27 - Porcello (2)
14 May 31 - Sale (2) (L5)
15 June 2 - Price (4)
16 June 4 - Sale (3)
17 June 8 - Price (5)
18 June 13 - Price (6)
19 June 14 - Erod (3)
20 June 15 - Sale (4)
21 June 18 - Price (7) (L6)
22 June 25 - Price (8)
23 June 26 - Sale (5) (L7)
24 June 30 - Erod (4) (L8)
25 July 2 - Price (9)
26 July 4 - Velazquez (4)
27 July 5 - Erod (5)
28 July 6 - Porcello (3)
29 July 7 - Price (10)
30 July 16 - Cashner (1) (L9)
31 July 17 - Erod (6)
32 July 18 - Sale (6)
33 July 20 - Porcello (4)
34 July 26 - Cashner (2)
35 July 27 - Erod (7)
36 Aug 2 - Erod (8) (L10)
37 Aug 4 - Price (11) (L11)
38 Aug 5 - Porcello (5)
39 Aug 6 - Cashner (3) (L12)
40 Aug 11 - Cashner (4) (L13)
41 Aug 16 - Porcello (6)
42 Aug 17 - Erod (9)
43 Aug 18 - Eovaldi (2)
44 Aug 21 - Porcello (7)
Interesting: 43 times this season we have started a pitcher having a better season to date than the opposition's pitcher (ERA, but whatever, see above.) We went 31-13 in those games. Although it must be said Cashner garnered 4 starts and 4 losses.
Price faced a pitcher having a worse season than he did 11 times. Erod 9, Porcello 7, Sale 6, Cashner 4.